Group actions
and the enumeration of states
in condensed matter physics
Wojciech Florek, Adalbert Kerber,
Barbara Lulek, Tadeusz Lulek
April 19, 2006
Contents
Introduction
1
1 Classes of configurations
5
1.1
Actions of groups on sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
1.2
Orbits and double cosets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
1.3
Classes of configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
1.4
Finite symmetric groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
1.5
Complete monomial groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
1.6
Hidden symmetries and fibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
1.7
Graphs and interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
2 Enumeration of configuration classes
47
2.1
Enumeration of configuration classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
2.2
The Involution Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
2.3
Special symmetry classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
2.4
Counting by stabilizer class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
2.5
Tables of marks and Burnside matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
2.6
Actions on posets, semigroups, lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
3 Weights
81
3.1
Enumeration by weight
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
82
3.2
Cycle indicator polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
3.3
Sums of cycle indicators, recursive methods . . . . . . . . . . . . . . . . . . . . . . . . . . . .
96
3.4
Weighted enumeration by stabilizer class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
3.5
The Burnside ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4 Classes of configurations, constructive aspects
117
4.1
The evaluation of classes of configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
4.2
Transversals of configuration classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
4.3
Orbits of centralizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
4.4
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
4.5
Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
4.6
Recursive construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
4.7
Orderly generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
4.8
Generating orbit representatives
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
i
Introduction
A prominent object of mathematical interest is the set
Y X := {f : X → Y }
consisting of all the mappings f : X → Y from X into Y , where both X and Y are given sets. There are many
objects of this form in physics, too, since X may be considered as being a set of places, positions, boxes
or alike, and Y can be a set of colours, spin values, and so on. An element f ∈ Y X then may be called a
configuration of the set X, with values in the set Y , and so the set Y X is the set of all the configurations of
this particular form.
We shall introduce various structures on the set Y X , mainly by group actions, in order to classify the
configurations by introducing notions that allow to specify which of the configurations are essentially different
in the case when a symmetry group is in the game.
In this way we shall demonstrate in the present book applications of classification of configurations and of
a series of related mathematical structures to a variety of physical theories and models, with the emphasize
on physics of condensed matter. This will be done by suitable choices of sets X and Y and corresponding
actions of groups on these sets which introduce actions on Y X in a canonic way.
The most general group acting on the set X is the symmetric group SX , i.e. the group of all permutations
on X. Taking also the symmetric group SY and various subgroups G ⊆ SX and H ⊆ SY into account,
we shall obtain natural actions on Y X . The corresponding mathematical theory is the famous Pólya theory
of enumeration, by which we can count the number of essentially different configurations with respect to
a given symmetry group. Of course, the counting methods will be refined in order to be able to construct
representatives of the classes of configurations and in order to put the hands on these classes and directly
to see what these classes are.
The main tool is to introduce problem oriented actions of groups G and H, their direct product H ×G, and
their wreath product H o G on Y X . Such actions provide a rich permutational structure on the set Y X ,
since they define a hierarchy of classifications of configurations. It is expressed in terms of orbits, strata,
stabilizers, fixed points, epikernels, etc.
This method can be nicely demonstrated in the case of the Heisenberg model of a magnet, which we choose
as a paradigmatic example of physical applications throughout the book. Here X is the set of all magnetic
nodes of a crystal, whereas Y serves as the set of all magnetic quantum numbers m of the single-node
spin s (so that |Y | = 2s + 1). The groups of interest in this case are primarily those, which describe the
symmetry of the model, e.g. the group G ⊆ SX of all those permutations of nodes imposed by the geometric
symmetry of the crystal, or the two-element time-reversal group H ⊆ SY , which is responsible for the change
of sign of the magnetic quantum number m. Actions of these groups carry some exact quantum numbers,
which immediately yield an information about the system. But also some other groups are in the interest of
physicists since they provide useful classification labels, or, in other words, a complete set of initial quantum
numbers for appropriate calculations of energy spectra and stationary states of the system. In particular, the
symmetric group SX itself, which in general is not the symmetry group of the system, yields a classification
scheme including the total magnetization M — the exact quantum number invariant under SX . Thus, the
permutational structure of the set Y X of all magnetic configurations reveals such intrinsic properties of this
set as the list of types and numbers of orbits with a given total magnetization.
The purely combinatorial picture of actions of groups on sets finds its natural extension in cases when the
given set is equipped with an additional structure. A particularly important situation in physics is that of
1
2
INTRODUCTION
an additional vector space (or, more generally, of a module) structure. A set X can be easily equipped with
the structure of a linear space over a field F by taking all linear combinations of set elements with coefficients
in F. And again, this is a set of the form Y X , namely
FX .
In physics, one most frequently meets the field R of real numbers (e.g. the space of velocities relative to a
given inertial frame; this space can be spanned by any set X of order three, and so we usually write R3 for
it) and the field C of complex numbers (e.g. a Hilbert space of quantum states of a physical system, spanned
by the set X of quantum numbers sequences; each sequence is the set of eigenvalues of a complete set of
commuting operators, as it was introduced by Dirac1 ). But sometimes one can also find other fields like that
of rational numbers Q (non-primitive translations in crystallography), cyclotomic fields (incommensurate
crystals), or finite and p-adic fields (symmetries of polymer chains). A crystal lattice in n dimensions serves
as an example of an n-dimensional module over the ring Z of integers, whereas a crystal with the Born–von
Kármán periodic boundary conditions (with the same period N in each direction) is a module over the ring
ZN of residues modulo N . In particular, for prime N = p, it becomes the n-dimensional space over the finite
field Zp .
Permutational action of a group G on a set X has now to be extended to the linear action on the space or
on the module. The permutational classification is not affected by the linear extension since each subspace
spanned by an orbit remains invariant, but we get finer results due to the notions of decomposability and
irreducibility. In the linear case, invariant subspaces spanned by orbits decompose into more elementary
subspaces, which are irreducible or indecomposable.
It is worthwhile to point out here that the linear structure is nowadays a natural feature of most of physical
theories of condensed matter. It is due to the fact that these theories emerge from quantum mechanics, and
thus they have to fulfil its basic principle of the linear superposition of states. It follows that any finite or
discrete set of quantum states Y X has to be embedded properly into an appropriate linear space. In the
particular case of a spin system one way of doing this is simply to take the set Y of magnetic quantum
numbers as orthogonal basis of the (2s+1)-dimensional unitary linear space
CY
of eigenstates of the single-node spin with the fixed value s, and then this basis has been chosen using
appropriate Racah conventions concerning phases and Hermitian conjugation (time reversal), as described
in textbooks of quantum mechanics.
Having imposed the linear structure, one can further introduce a metric tensor which yields various types
of spaces: Euclidean (e.g. the space of positions of a free particle), pseudo-Euclidean (e.g. the Minkowski space-time), or unitary (Hilbert spaces of quantum mechanics). This requirement does not affect the
classification scheme, but admits only orthonormal bases, and thus orthogonal or unitary forms of irreducible
representations are required.
In particular, the unitary structure is related to Weyl duality between the symmetric and unitary groups,
which he put as a basis for a quantum mechanical description of the system of N identical particles, just in
the ‘pioneer’ period of theory of quanta. Let L be a space of quantum states of a single particle. Then, all
the states of the system of N particles, each confined to the space L, are some (not all!) elements of the
N −th tensor power space
LN := L ⊗ L ⊗ . . . ⊗ L .
{z
}
|
N times
N
The space L is the central object of the Weyl duality. It admits the permutational action of the symmetric
group SN , which permutes the N factors of the tensor power LN , and the simultaneous linear action generated
by the unitary group U (n) on the single-particle spaces L, where dim L = n. Both actions commute, they
are completely reducible, and an associated decomposition into irreducible subspaces provides a complete
classification of the corresponding orthonormal basis states in the tensor power space LN . In other words,
both actions taken together suffice to provide a complete classification, without extra labels for repeated
1 In fact, we demonstrate in this book that group actions can serve as tools to propose such complete sets of labels for
symmetry adapted bases. In other words, we propose a complete classification of basic states in a space by means of group
actions.
INTRODUCTION
3
irreducible representations of both groups. Thus, the duality of Weyl between the actions of the symmetric
and unitary groups in the tensor power space provides a powerful tool for classification of states.
In the context of the present book, the tensor power space LN can be interpreted as a linear extension of
the set Y X of all configurations. It can be done by identifying the sets X and Y with the corresponding sets
of labels
X = {1, 2, . . . , N }, Y = {1, 2, . . . , n},
and L being the linear closure of Y , i.e. the set of mappings
L = CY .
This yields in fact the isomorphism
LN ' C(Y
X
)
.
In this way, the permutational structure of the set of all configurations finds its reflection within the picture
of decomposition of the tensor power space into irreducible subspaces.
The aim of the book is to put some combinatorial, algebraic and group-theoretic constructions related to
subjects discussed above in the context of examples from physics. We believe that the book can be useful
for physicists, mainly due to demonstration of applications of some mathematical methods in physics. We
also hope that mathematicians can find here a reservoir of examples, how various mathematical structures
— seemingly quite abstract and far from reality — are in fact useful as surprisingly adequate tools for many
descriptions of models of physical systems and processes envolved therein.
On the side of the reader we expect basic knowledge of linear algebra and algebra, but most of the terms
are defined and described ‘from scratch’.
Finally we should like to express our sincerest thanks for scientific support to many colleagues, in particular
to the participants of the seminars on “Symmetry and Structural Properties of Condensed Matter” held at
Zaja̧czkowo repeatedly, and for financial support to the DAAD.
file:f01.tex
4
INTRODUCTION
Chapter 1
Classes of configurations
This introductory chapter starts with a description of the basic notions of the theory of actions of groups:
orbits, stabilizers, fixed points, and so on. The Cauchy-Frobenius Lemma is derived, which yields the number
of orbits in the case when both the group and the set on which it acts are finite. In order to prepare the
applications of this lemma and its refinements which follow in the next chapters, a detailed description of
the conjugacy classes of symmetric and of monomial groups is added. We also present cyclic, dihedral, and
alternating subgroups of symmetric groups.
The paradigmatic actions which we discuss and apply in full detail here and later on are several natural actions
on the set Y X , consisting of all the mappings from X into Y . These mappings are called configurations, X
can be considered as a set of nodes (say, of atoms in a crystal), while Y can be interpreted as a set of colors
(spin projections, say). The actions on Y X are induced in a natural way by actions of symmetry groups on
X and/or on Y . The corresponding orbits are called classes of configurations and there are many structures
in physics, mathematics and chemistry, which can be defined as classes of this kind. For example, in the last
section of this chapter we show that graphs can be considered as such classes.
A very simple but also very illustrative example of the structures that we are going to discuss is presented in
figure 1.1, which shows the 8 essentially different ways of coloring the vertices of a regular pentagon in the
two colors • and ◦: Essentially different means that no symmetry operation applied to the pentagon leads
from one coloring to another one.
This example seems to be only a plaything but, in fact, it shows the main features of the problems considered
in this book. First of all, there is a set X, in the example it is the set of vertices of the pentagon. Secondly,
there is a set Y , consisting of the two colors. And a coloring is a mapping from the set X into the set Y ,
i.e. it is an element of the set
Y X := {f : X → Y }.
But there are, of course, altogether 25 = 32 such colorings, not all of them being essentially different, since
there is the symmetry group of the pentagon, consisting of rotations and reflections, which, by application
to the pentagon, may interchange colorings.
In mathematical terms: the symmetry group of the pentagon acts on the set of colorings, and it induces
its decomposition into exactly 8 pairwise disjoint subsets, representatives of which are shown in the picture.
c
Z
Zc
c
B
Bc
c
s
Z
s Zs
B
Bs
s
s
Z
Zc
c
B
Bc
c
c
Z
Zs
s
B
s
Bs
s
Z
c Zs
B
Bc
c
c
Z
s Zc
B
Bs
s
s
Z
Zc
c
B
Bc
s
c
Z
Zs
s
B
c
Bs
Figure 1.1: The essentially different colorings of a pentagon
5
6
Chapter 1. Classes of configurations
These subsets will be called orbits later on. Summarizing, the main features of the structures we are discussing
here are the following ones:
• They can be defined as orbits of finite groups (symmetry groups) on sets of mappings between finite
sets.
• Having defined the structures as orbits, we can enumerate them, and we can refine this enumeration.
• Finally we can quite often construct a system of representatives of the orbits in order to put our hands
on them. In small cases we can even generate a complete catalog of them.
There are many similar situations in physics. The set X of objects may consist of vertices, balls, particles,
atoms, nodes etc., and the set Y may consist of colors or other properties the elements of X may have,
for example electric charges, valences or simply natural or integral numbers which may indicate, say, spin
projections. Thus the set Y X of all the mappings from X to Y , which we call the set of configurations, is
of great interest. If there is a symmetry group in the game, the question arises, which of the configurations
are essentially different. A symmetry group comes in if we impose restrictions on X: geometric, physical,
chemical ones etc. For example, the geometric restriction that the five vertices are the vertices of a regular
pentagon, leads to the symmetry group G, say, of the pentagon, and therefore, not all the 32 colorings are
essentially different. In mathematical terms, the symmetry group G acts on X and this action induces an
action of G on the set of configurations. For example, the symmetry group of the pentagon induces an
action on the set of the 32 colorings of the pentagon, and this action forms eight orbits and a system of
representatives of these eight orbits is shown in the picture given above.
Moreover, there may also be a symmetry group H, say, of the set Y . For example, if we assume the two
colors to be interchangeable, then it is not only G or H that acts on Y X , it is even their direct product
H × G that acts on Y X . For example, in the case of the colorings, H × G is the group {1, τ } × G, where
again G is the symmetry group of the pentagon, and τ is the transposition ◦ ↔ •. It is clear from the above
picture, that this bigger group has exactly 4 orbits on the set of colorings, since the second row of colorings
can be obtained from the first one under the transposition τ . Here are further examples of this particular
form.
Systems of identical particles Let X be a set of identical particles, and let Y be the set of single-particle
states. E.g., in the theory of nucleons the set Y = {n, p} consists of two elements: neutron (n) and proton
(p) states of a nucleon. Similarly, in the theory of quarks the set Y consists of three “colors” of quarks.
Theories of elementary particles, in particular quantum chromodynamics, deal with configurations f : X → Y
as intermediate constructions in determination of states of an assembly X of particles. One has to stress,
however, that — besides special cases — a single configuration does not describe a real physical system for
reason of the so called “superselection rules” imposed by the principle of indistinguishability of identical
particles.
Coherent radiation Consider a system X consisting of n identical molecules, with m energy levels in a
set Y , together with the radiation field, confined to a cavity (a model of laser). Then each configuration
f : X → Y corresponds to a particular energy of the subsystem of all molecules (the “matter”), and the set
Y X of all configurations serves as a step in construction of states of the whole system matter–radiation. In
particular, some specific linear combinations of configurations yield superradiant states, responsible for laser
beam. (The fact that linear combinations occur shows that besides group actions we shall use representations,
which means group actions which are linear!)
Our main aim, however, is to present possible applications of group actions in solid state physics. As the
paradigmatic example we have chosen the Heisenberg model of magnetism, but structures and methods are
introduced in a general way, so it is easy to apply them to any similar model. In the following sections and
chapters we describe, in terms of group actions, what means a crystal lattice, a pair of nearest-neighbors
etc., and we introduce many other structures, which will lead us to adequate mathematical description of a
problem under consideration. Having done this we will be able to study and to determine physical properties
of the assumed model.
file:f10.tex
1.1. Actions of groups on sets
1.1
7
Actions of groups on sets
Let G denote a multiplicative group and X a nonempty set. An action of G on X is described by a mapping
G × X → X: (g, x) 7→ gx, such that g(g 0 x) = (gg 0 )x, and 1x = x.
We abbreviate this by saying that G acts on X or simply by calling X a G–set or by writing
G X,
in short, since G acts from the left on X. (Actions from the right can be introduced analogously.) Before
we provide more details and examples, we mention a second but equivalent formulation. A homomorphism
δ from G into the symmetric group
SX := {π | π: X → X, bijectively}
on X is called a permutation representation of G on X. It is easy to check (exercise 1.1.1) that the definition
of action given above is equivalent to
δ: G → SX : g 7→ ḡ, where ḡ: x 7→ gx, is a permutation representation.
b and so we have, if Ḡ := δ[G], the isomorphism
The kernel of δ will be denoted by G,
1.1.1
b
Ḡ ' G/G.
ov
b = {1}, i.e. when gx = x for all x if and only if g = 1, the action is said to be faithful.
In the case when G
On the other side we have the trivial action, i.e. when gx = x for all x ∈ X and all g ∈ G or, in other
b = G. Note that this is the unique action if G = {1} or X = {x}. Another simple example is
words, G
the natural action of SX on X itself, where the corresponding permutation representation δ: π 7→ π̄ is the
identity mapping. A number of less trivial examples will follow in a moment.
When both G and X are finite, we call the action a finite action. We notice that, according to 1.1.1, for
each finite G–set X, we may also assume without loss of generality that G is finite. This case, i.e. a finite
action, will be considered in this book since we are interested in enumeration problems. However, the basic
ideas can be also applied for infinite sets X and infinite groups G.
The definition of an action by the homomorphism δ suggests that there may exist many (sometimes infinitely
many) different actions for a given group G and a given set X. This follows from the fact that there may
be many possible homomorphisms G → SX . On the other hand, there are cases in which only the trivial
action can be defined, because there are no nontrivial homomorphisms G → SX . It follows from Lagrange’s
Theorem (see below) that the order of Ḡ divides both the order of G and the order of SX . Consequently,
it is impossible, for example, to define a nontrivial action of 5-element group on a 3-element set since
gcd(5, 3!) = 1.
An action of G on X has first of all the following property which is immediate from the two conditions
mentioned in its definition:
1.1.2
gx = x0 ⇐⇒ x = g −1 x0 .
00
This is the reason for the fact that G X induces several structures on X and G, and it is the close arithmetic
and algebraic connection between these structures which makes the concept of group action so efficient. First
of all the action induces the following equivalence relation on X (exercise 1.1.2):
1.1.3
x ∼G x0 : ⇐⇒ ∃ g ∈ G: x0 = gx.
The equivalence classes are called orbits, and the orbit of x ∈ X will be indicated as follows:
G(x) : = {gx | g ∈ G}.
00
8
Chapter 1. Classes of configurations
As ∼G is an equivalence relation on X, a transversal, which means a system of representatives T of the
orbits, yields a set partition of X, i.e. a complete dissection of X into the pairwise disjoint and nonempty
subsets G(t), t ∈ T :
[
˙
pa
1.1.4
X=
G(t).
t∈T
The set of all orbits will be denoted by
G\\X : = {G(t) | t ∈ T }.
If G has exactly one orbit on X, i.e. if and only if G\\X = {X}, then we say that the action is transitive, or
that G acts transitively on X.
According to 1.1.4 an action of G on X yields a partition of X. It is trivial but very important to notice
that also the converse is true: Each set partition of X gives rise to an action of a certain group G on X
as follows. Let I be an index set and let Xi , i ∈ I, denote the blocks of the set partition in question, i.e.
the Xi are nonempty, pairwise disjoint, and their union is equal to X. Then the following subgroup of the
symmetric group SX acts in a natural way on X and has the Xi as its orbits:
young
1.1.5
M
SXi := {π ∈ SX | ∀ i ∈ I: πXi = Xi }.
i
(We note in passing, that usually there are several groups that have these orbits, for example, we can replace
the symmetric group SXi by a cyclic one.) Groups of this form are called Young subgroups, since they also
play a crucial rôle in the representation theory of the symmetric group (see below) which owes very much
to Alfred Young.
Summarizing our considerations in two sentences, we have obtained:
1.1.6 Corollary An action of a group G on a set X is equivalent to a permutation representation of G
on X and it yields a set partition of X into orbits. Conversely, each set partition of X corresponds in a
natural way to an action of a certain Young subgroup of the symmetric group SX which has the blocks of the
partition as its orbits.
To the orbits G(x), which are subsets of X, there correspond certain subgroups of G. For each x ∈ X we
introduce its stabilizer :
Gx : = {g | gx = x}.
There are also stabilizers of subsets M ⊆ G, and here we distinguish if their elements stabilize the set M
pointwise or setwise. The pointwise stabilizer, which is also called the centralizer of M will be denoted by
GM : = {g | ∀ m ∈ M : gm = m}.
The setwise stabilizer, which is also called the normalizer of M, will be indicated by
G{M } : = {g | ∀ m ∈ M : gm ∈ M }.
The last one of the fundamental concepts induced by an action of G on X is that of fixed points. A point
x ∈ X is said to be fixed under g ∈ G if and only if gx = x, and the set of all the fixed points of g is indicated
by
Xg : = {x | gx = x}.
More generally, for any subset S ⊆ G, we put
XS : = {x | ∀g ∈ S: gx = x}.
The particular case XG is called the set of invariants.
The first bunch of examples which illustrate these concepts will show that various important group theoretical
structures can be considered as orbits or stabilizers. We mention these structures since they will occur quite
often later on:
1.1. Actions of groups on sets
ex1
9
1.1.7 Examples If G denotes a group and U denotes its subgroup (in short: U ≤ G), then
• G acts on itself by left multiplication: G × G → G : (g, x) 7→ g · x. This action is called the (left)
regular representation of G, it is obviously transitive, and all the stabilizers are equal to the identity
subgroup {1}. (See also exercise 1.1.3.)
• G acts on itself by conjugation: G × G → G: (g, x) 7→ g · x · g −1 . The orbits of this action are the
conjugacy classes of elements, and the stabilizers are the centralizers of elements (in accordance with
the above definition of centralizer of a subset):
G(x) = C G (x) := {gxg −1 | g ∈ G},
and
Gx = CG (x) := {g | gxg −1 = x}.
• U acts on G by left multiplication:
U × G → G: (u, g) 7→ ug.
This is in fact a restriction
cosets:
UG
of the (left) regular representation
GG
and its orbits are the right
U (g) = U g := {ug | u ∈ U }.
There is a corresponding action of U from the right on G, its orbits are the left cosets:
gU := {gu | u ∈ U }.
In both cases stabilizers are equal to {1}, of course.
• G acts on the set G/U := {xU | x ∈ G} of left cosets as follows:
G × G/U → G/U : (g, xU ) 7→ gxU.
This action is transitive, and the stabilizer of xU is the subgroup xU x−1 which is conjugate to U .
• G acts on the set L(G) := {U | U ≤ G} (the lattice) of all its subgroups by conjugation:
G × L(G) → L(G): (g, U ) 7→ gU g −1 .
The orbits of this action are the conjugacy classes of subgroups, and the stabilizers are the normalizers
(again in accordance with the above definition of normalizer of a subset):
e := { gU g −1 | g ∈ G},
G(U ) = U
and
GU = NG (U ) := {g | gU = Ug}.
3
Returning to the general case we first state the main (and obvious) property of the stabilizers of elements
belonging to the same orbit:
1.1.8
fx = {gGx g −1 | g ∈ G} = {Gx0 | x0 ∈ G(x)}.
Ggx = gGx g −1 , G
This shows in particular, that the stabilizers of elements in the same orbit are conjugate subgroups, and,
moreover, that they form a complete class of conjugate subgroups. For example, if the stabilizer of an
element is equal to {1} then this is true for all the elements in that orbit. These orbits are sometimes of
great interest, and therefore we shall give them a special name by calling them regular or asymmetric orbits
of G. A transitive action with a regular orbit is called a regular action.
But the crucial point is the following natural bijection between the orbit of x and the set of left cosets of its
stabilizer:
or
10
Chapter 1. Classes of configurations
1.1.9 Lemma The mapping G(x) → G/Gx : gx 7→ gGx is a bijection.
bi
This result shows in particular that the length of the orbit is the index of the stabilizer. Thus, if G is a finite
group acting on the set X, then for each x ∈ X we have
in
|G(x)| = |G|/|Gx |.
1.1.10
This arithmetic relation between the length of an orbit and the order of the stabilizer of any of its elements
is a very strong condition. It implies various useful identities. For example, considering the action of G on
G/U one obtains
|G(U x)| = |G|/|xU x−1 | = |G|/|U |,
since conjugated subgroups have the same order. This means that |U |, the order of the subgroup, divides
the order |G| of G (Lagrange’s Theorem). The previous example (i.e. the action U G) leads to the following
formula:
|U g| = |U (g)| = |U |,
since for all g ∈ G we have Ug = {1}. Two other important cases directly follow from the actions by
conjugation on elements and subgroups: If G is finite, g ∈ G, and U ≤ G, then the orders of the conjugacy
classes of elements and of subgroups satisfy the following equations:
classorder
1.1.11
e | = |G|/|NG (U )|.
|C G (g)| = |G|/|CG (g)| and |U
The result in 1.1.10 is very important, it is essential in the proof of the following counting lemma which,
together with later refinements, forms the basic tool of the theory of enumeration under finite group action:
CF
1.1.12 The Lemma of Cauchy-Frobenius The number of orbits of a finite group G acting on a finite
set X is equal to the average number of fixed points:
|G\\X| =
1 X
|Xg |.
|G|
g∈G
Proof:
X
g∈G
|Xg | =
X X
g
x∈Xg
1=
X X
1=
x g∈Gx
X
|Gx |,
x
which is, by 1.1.10, equal to
X |G|
=
|G(x)|
x
X X
ω∈G\\X x∈ω
|G|
= |G\\X|.
|G(x)|
2
The next remark helps considerably to shorten the calculations necessary for applications of this lemma. It
shows that we can replace the summation over all g ∈ G by a summation over a transversal of the conjugacy
classes, as the number of fixed points turns out to be constant on each such class:
1111
1.1.13 Lemma The mapping
Xg0 → Xgg0 g−1 : x 7→ gx
is a bijection, and hence
χ: G → N: g 7→ |Xg |
is a class function, i.e. it is constant on the conjugacy classes of G. More formally, for any g, g 0 ∈ G, we
have that |Xg0 | = |Xgg0 g−1 |.
The mapping χ is called the character of the action of G on X. Applying this to 1.1.12 we obtain
1.1. Actions of groups on sets
1112
1.1.14 Corollary Let
Then
GX
11
be a finite action and let C denote a transversal of the conjugacy classes of G.
1 X G
|C (g)| |Xg |
|G|
g∈C
X
X
=
|CG (g)|−1 |Xg | =
|CG (g)|−1 χ(g).
|G\\X| =
g∈C
g∈C
Another formulation of the Cauchy-Frobenius Lemma makes use of the permutation representation g 7→ ḡ
defined by the action in question. The permutation group Ḡ, being the image of G under this representation,
yields the action Ḡ X of Ḡ on X, which has the same orbits, and so we also have:
1.1.15 Corollary If X is a finite G–set, then (for any group G) the following identity holds:
|G\\X| =
11
1 X Ḡ
1 X
|Xḡ | =
|C (ḡ)||Xḡ |,
|Ḡ|
|Ḡ|
ḡ∈C̄
ḡ∈Ḡ
where C¯ denotes a transversal of the conjugacy classes of Ḡ.
Under enumerative aspects G X is essentially the same as Ḡ X. This leads to the question of a suitable
concept of morphism between actions of groups. To begin with, two actions will be called isomorphic or
equivariant iff they differ only by an isomorphism η: G ' H of the groups and a bijection θ: X → Y between
the sets which satisfy η(g)θ(x) = θ(gx). In this case we shall write
GX
'
H Y,
in order to indicate the existence of such a pair of mappings. If G = H we call G X and G Y similar actions,
if and only if they are isomorphic by (η, θ) and, moreover, η = id G (the identity mapping; cf. exercise 1.1.7).
We indicate this by
G X ≈ G Y.
Important special cases follow directly from the proof of 1.1.9:
1.1.16 Lemma If
GX
is transitive then, for any x ∈ X, we have
GX
≈
tr
G (G/Gx ).
A weaker concept is that of G-homomorphy. We shall write
GX
∼ GY
if and only if there exists a mapping θ: X → Y which is compatible with the action of G: θ(gx) = gθ(x).
Later on we shall see that the use of G-homomorphisms is one of the most important tools in the constructive
theory of discrete structures which can be defined as orbits of groups on finite sets. A characterization of
G-homomorphy gives
1.1.17 Lemma Two actions
y ∈ Y such that Gx ⊆ Gy .
GX
and
GY
are G-homomorphic if and only if, for each x ∈ X, there exist G
When the relation η(g)θ(x) = θ(gx) holds for an epimorphism η: G → H and θ to be a mapping θ: X → Y ,
then we call (η, θ) an epimorphism of actions. (Note that any homomorphism φ: G → H defines, according to
the Homomorphism Theorem, an epimorphism φ: G → im φ.) The special case when η: G → id Y is a trivial
homomorphism yields θ(x) = θ(gx) for all x ∈ X and all g ∈ G, i.e. the question of mappings constant on
orbits G\\X arises. For example, the character of an action is such a mapping for the action of G on itself
by conjugation (see 1.1.7).
An application of lemma 1.1.16 to a regular orbit leads immediately to
12
1.1.18 Corollary If
Chapter 1. Classes of configurations
GX
is regular, i.e. if each Gx = {1}, then we have
GX
≈
re
G G.
This similarity means in particular, that the elements of a regular orbit X can be labelled by the elements of
the group G. After choosing a representative x1 the other elements x ∈ X can be labelled as xg in accordance
with the bijection 1.1.9, i.e. xg := gx1 . However, it must be stressed that such a labelling scheme depends
on the choice of x1 .
crystal
1.1.19 Crystal lattice A special case of such a construction is an affine space A over a vector space V ,
wich can be defined as a regular orbit of the additive group of vectors in V , i.e. for any a ∈ A and v, w ∈ V
we have (v + w) + a = v + (w + a) and, moreover,
v + a = a ⇐⇒ v = 0 ∈ V
(since V is an abelian group we use, due to tradition, the additive notation for both the addition of vectors
and the action V on A). This means that the action V A (the addition of vectors to points) is similar to the
left regular action V V (the addition of vectors) and choosing one point, a0 say, we can label the others by
v ∈ V , i.e. av := v + a0 .
Let us assume now that V is the n–dimensional space over the field of real numbers with a basis B = {bi |
1 ≤ i ≤ n} and introduce a subgroup T (not subspace!) of V defined as
T := {v ∈ V | v =
n
X
ti bi , ti ∈ Z}.
i=1
The restriction T A of the action V A leads to a decomposition of A into regular orbits of T , so points
a ∈ T (a0 ) can now be labelled by n-tuples (t1 , . . . , tn ) of integers. This construction enables us to define the
n-dimensional crystal lattice Λ as a regular orbit of the translation group T < V . Elements of this orbit are
called crystal nodes. Note that T , and therefore also a crystal lattice, depends of our choice of the basis B
and we should rather write T (B) and Λ(B). Since T < V and Λ < A we may consider restrictions to T and
to Λ of any other structures V is equipped with, e.g. a scalar product leading to the notion of orthogonality.
3
Remark Note that a set X under consideration may have additional structure: it can be a group, a vector
space, a graph, a molecule, a set of mappings etc. If this ‘intrinsic’ structure is important in the investigated
problem then the action of the group G may respect that additional structure. For example, when X is an
algebraic structure we can consider its automorphism group Aut X (and/or its subgroups) with the action
defined in a natural way as
Aut X × X → X: (ψ, x) 7→ ψ(x).
Such actions respect the intrinsic structure. The group Aut X is called the symmetry group of X, and each
subgroup G of it is called a symmetry group or a group of symmetries. In such cases one may require θ to
be not only a bijection but also a homomorphism of the appropriate (algebraic) structures. Therefore it is
quite often possible to construct two actions which are similar while θ: X → Y is not an isomorphism or a
homomorphism.
The case of X being a vector space, when Aut X consits of linear mappings, is very important in physical
applications. This situation will be discussed carefully later on. Among other things, in linear representation
theory we shall use a stronger concept than above.
3
Exercises
E111
E 1.1.1 Assume X to be a G–set and check carefully that g 7→ ḡ is in fact a permutation representation,
i.e. that ḡ ∈ SX and that g1 · g2 = g1 · g2 .
1.1. Actions of groups on sets
13
Verify 1.1.2 and prove that ∼G defined in 1.1.3 is in fact an equivalence relation.
E112
E 1.1.2
E113
E 1.1.3 Check that the (right) regular representation of G, i.e. if we want G to act on itself by right
multiplication, has to be defined as (g, x) 7→ x · g −1 .
E 1.1.4
Prove 1.1.9 and 1.1.12.
E 1.1.5
Let
GX
be finite and transitive. Consider an arbitrary x ∈ X and prove that
|Gx \\X| =
E
1 X
|Xg |2 .
|G|
g∈G
E 1.1.6 Check that the G-isomorphy ' (and hence also the G-similarity ≈) is an equivalence relation on E
group actions.
E 1.1.7 Consider the following definition: We call actions G X and G Y inner isomorphic if and only if E
there exists a pair (η, θ) such that G X ' G Y and where η is an inner automorphism, which means that
η: G → G: g 7→ g 0 gg 0−1 ,
for a suitable g 0 ∈ G. Show that this equivalence relation has the same classes as ≈.
file:f11.tex
14
Chapter 1. Classes of configurations
1.2
Orbits and double cosets
From a given action we can derive various other actions in a natural way, e.g. G X yields Ḡ X, Ḡ being the
homomorphic image of G in SX , which was already mentioned. We also obtain the subactions G M on subsets
M ⊆ X which are nonempty unions of orbits. Furthermore there are the restrictions U X to the subgroups U
of G. As the orbits of G X are unions of orbits of U X, the comparison of actions and restrictions is a suitable
way of generalizing or specializing structures if they can be defined as orbits. The following example will
show what is meant by this (cf. also exercise 1.2.1).
ex121
1.2.1 Example Let U denote a subgroup of the direct product G × G. Then U acts on G as follows (cf.
also exercise 1.1.3):
U × G → G : ((a, b), g) 7→ agb−1 .
The orbits U (g) = {agb−1 | (a, b) ∈ U } of this action are called the bilateral classes of G with respect to
U . By specializing U we obtain various interesting group theoretical structures some of which have been
mentioned already (cf. 1.1.7):
• If A is a subgroup of G, then both A × {1} and {1} × A are subgroups of G × G. Their orbits are the
subsets
(A × {1})(g) = Ag,
the right cosets of A in G, and
({1} × A)(g) = gA,
the left cosets of A in G.
• Another subgroup of G × G is its diagonal subgroup
∆(G × G) := {(g, g) | g ∈ G}
(which is isomorphic with G). Its orbits are the conjugacy classes:
∆(G × G)(g) = {g 0 gg 0−1 | g 0 ∈ G} = C G (g).
• If B denotes a second subgroup of G, then we can put U equal to the subgroup A × B, obtaining as
orbits the (A, B)–double cosets of G:
(A × B)(g) = AgB.
3
Hence left and right cosets, conjugacy classes and double cosets turn out to be special cases of bilateral classes.
Being orbits, two of them are either equal or disjoint, moreover, their order is the index of the stabilizer of an
element. We have mentioned this already in connection with conjugacy classes and centralizers of elements.
Here is the consequence for double cosets: Since the stabilizer of g in A × B satisfies
(A × B)g = {(gbg −1 , b) ∈ A × B | b ∈ B} = A ∩ gBg −1 ,
we obtain
cdequation
1.2.2 Corollary If G denotes a finite group with subgroups A and B, then
|AgB| =
|A| |B|
,
|A ∩ gBg −1 |
and if D denotes a transversal of the set A\G/B of (A, B)-double cosets, then
|G| =
X
g∈D
|AgB| =
X
g∈D
|A| |B|
.
|A ∩ gBg −1 |
2
1.2. Orbits and double cosets
15
The Cauchy-Frobenius Lemma yields the number of bilateral classes. In order to evaluate it we have to
calculate the number |G(a,b) | of fixed points for a given (a, b) ∈ U, which is
|{g | a = gbg −1 }| =
n
|CG (a)| = |CG (b)| if a, b are conjugates,
0
otherwise.
Thus, by the Cauchy-Frobenius Lemma, we obtain
|U \\G| =
1.2.3
|G| X |C G (a) ∩ C G (b)|
.
|U |
|C G (a)|2
ul
(a,b)∈U
1.2.4 Corollary If U denotes a subgroup of G × G, G being a finite group, then the number of bilateral bi
classes of G with respect to U is
|G| X |C G (g) × C G (g) ∩ U |
,
|U |
|C G (g)|
g∈C
where C denotes a transversal of the conjugacy classes of elements in G. In particular, the set
A\G/B := {AgB | g ∈ G} = (A × B)\\G
of (A, B)-double cosets has the order
|A\G/B| =
|G| X |C G (g) ∩ A| |C G (g) ∩ B|
.
|A| |B|
|C G (g)|
g∈C
2
1.2.5 Example Let us evaluate the number of double cosets of the group C6v (in the Schoenflies notation). db
This group is isomorphic to the dihedral group D6 and it consists of six rotations (C60 = E, C6 ,C62 = C3 ,
C63 = C2 , C64 = C3−1 , C65 = C6−1 ) and six reflections (σ0 , σ1 , σ2 , σ3 , σ4 , σ5 labelled in such a way that
σi = C6i σ0 for i = 1, . . . , 5). Let A = {E, σ2 } and B = {E, σ0 }. Since |C G (g) ∩ A| and |C G (g) ∩ B| are
non-zero only for g = E and g = σ0 (the latter one is a representative of the class {σ0 , σ2 , σ4 }) then corollary
1.2.4 gives
12
1
|A\G/B| =
(1 + ) = 4.
2·2
3
3
The main reason for the fact that double cosets show up nearly everywhere in the applications of group
actions is the following one (which we immediately obtain from 1.1.16):
1.2.6 Corollary If
natural bijection
GX
is transitive and U denotes a subgroup of G, then, for each x ∈ X we have the ub
ϕ: U \\X → U \G/Gx : U (gx) 7→ U gGx .
In particular, a transversal D of the set of double cosets U \G/Gx yields the following transversal of the set
of orbits of U :
T (U \\X) := {gx | g ∈ D}.
2
An application of this corollary to a regular orbit (cf. 1.1.18) leads to a trivial bijection
ϕ: U \\X → U \G: U (gx) 7→ U g.
16
Chapter 1. Classes of configurations
3 c
s
4T
Tc
2s
3 s
T
T1
σ0
s6
5
{C60 , σ0 }
3 s
4T
Ts
2 σ
2
T
c
Ts1
4T T
s
c6
5
{C61 , σ1 }
2c
3 c
T
Ts1
σ0
c6
5
{C63 , σ3 }
2 sσ
2
T
s
Tc1
4T T
s6
5
{C64 , σ4 }
σ4 2 s
T
T
Tc1
s
T
4T
T Tc
Ts6
5
T
3TT
{C62 , σ2 }
σ4 2 c
T
T
c T
Ts1
4T
T Ts
T6
5
T
3TTs
{C65 , σ5 }
Figure 1.2: An orbit of colorings of the 6–gon in three colors
1.2.7 Example Let us consider the set shown in figure 1.2, which is in fact an orbit of the action of the ex
group C6v ' D6 on all the 63 possible colorings of a regular 6-gon in 3 colors: “•”, “◦”and “ ”. Lines in
figure 1.2 denote reflections under which a given coloring is fixed. Below each element of the orbit we put
the corresponding left coset according to the bijection gx 7→ gGx (cf. 1.1.9). This action is transitive and,
according to 1.1.16, it is similar to the action of C6v on the left cosets determined by the stabilizer of an
element of this particular orbit. We use the stabilizer of the first element in the orbit, that is Gx = {E, σ0 }.
We are intersted in possible decompositions of this set into orbits under an action of a group U ≤ C6v .
Clearly the cyclic group is also transitive on this particular orbit, and so it does not decompose the orbit.
The other extreme is the identity subgroup which splits our set completely into one-element orbits. The
cyclic group generated by C2 and the cyclic group generated by C3 yield decompositions into 6/2 = 3 and
6/3 = 2 orbits, respectively.
The subgroup {E, σ2 } generated by the reflection σ2 fixes the second and the fifth coloring, whereas the
others form two 2-element orbits (the first and the third, the fourth and the sixth). This agrees with the
result presented in example 1.2.5 that there are four orbits. (See also exercise 1.2.4.)
3
dirsum
Now we take two actions into account, G X and H Y , say, and derive further actions from these. Without
loss of generality we can assume X ∩ Y = ∅ since otherwise we can rename the elements of X, in order to
˙ and let
replace G X by a similar action G X 0 , for which X 0 ∩ Y = ∅. Now we form the (disjoint) union X ∪Y
G × H act on this set as follows:
n
˙ ) → X ∪Y
˙ : ((g, h), z) 7→ gz if z ∈ X,
1.2.8
(G × H) × (X ∪Y
hz if z ∈ Y .
The corresponding permutation group will be denoted by Ḡ ⊕ H̄ (cf. 1.1.5) and called the direct sum of Ḡ
and H̄. Of course, this can easily be generalized to a direct sum of a finite number of actions. If G = H
then we can also consider a restriction of this action to the action of the diagonal subgroup ∆(G × G). Since
˙ ) in the following form:
this group is isomorphic to G we, in fact, obtain the action G (X ∪Y
disun-diag
1.2.9
˙ ) → X ∪Y
˙ : (g, z) 7→
G × (X ∪Y
gx if z = x ∈ X,
gy if z = y ∈ Y .
The special case of such an action is G X, when one writes X as a disjoint sum of orbits G(t) since (cf. 1.1.4)
[
[
˙
˙
G×
G(t) →
G(t): (g, x) 7→ gx
t∈T
t∈T
(x and gx belong to the same orbit G(t), of course). This description of G X suggests that each action can
be considered as a direct sum of actions G G(t), t ∈ T . The latter ones are transitive, and so each action is
in fact a direct sum of transitive actions.
1.2. Orbits and double cosets
17
Another canonic action of G × H is that on the cartesian product:
1.2.10
(G × H) × (X × Y ) → X × Y : ((g, h), (x, y)) 7→ (gx, hy).
ca
The corresponding permutation group will be denoted by Ḡ ⊗ H̄ and called the cartesian product of Ḡ and
H̄ (of course, we can again introduce a general case of the cartesian product of a finite number of groups
and sets). An important particular case is
1.2.11 Example Assume two finite and transitive actions of G on X and Y . They yield, as it was just ex
described, a canonic action of G × G on X × Y which has as one of its restrictions the action of ∆(G × G),
the diagonal, which is isomorphic to G, on X × Y . We notice that, for fixed x ∈ X, y ∈ Y , the following is
true (exercise 1.2.5):
• Each orbit of G on X × Y contains an element of the form (x, gy).
• The stabilizer of (x, gy) is Gx ∩ gGy g −1 , hence the action of G on the orbit of (x, gy) is similar to the
action of G on G/(Gx ∩ gGy g −1 ) (recall 1.1.16).
• (x, gy) lies in the orbit of (x, g 0 y) if and only if
Gx gGy = Gx g 0 Gy .
3
This proves the following:
1.2.12 Mackey’s Theorem If G acts transitively on both X and Y , then, for fixed x ∈ X, y ∈ Y , the tr
mapping
G\\(X × Y ) → Gx \G/Gy : G(x, gy) 7→ Gx gGy
is a bijection (note that G\\(X × Y ) stands for ∆(G × G)\\(X × Y )). Moreover, the action of G on the orbit
G(x, gy) and on the set of left cosets
G/(Gx ∩ gGy g −1 )
are similar. Hence, if D denotes a transversal of the set of double cosets Gx \G/Gy , for fixed x ∈ X, y ∈ Y ,
then we have the following similarity of actions of G :


[
 ˙ G/(Gx ∩ gGy g −1 ) .
G (X × Y ) ≈ G
g∈D
2
This theorem means that having two transitive actions (recall that each action can be written as a direct
sum of transitive actions) of G on sets (orbits) X and Y with stabilizers Gx and Gy , respectively, one can
construct the action of G on X × Y , which in general is not transitive, and decompose it into a direct sum
of transitive actions. The latter ones are labelled by representatives g ∈ D of double cosets Gx gGy and the
corresponding stabilizers are given as Gx ∩ gGy g −1 .
To be more concrete, let us suppose X = Y to be the set (orbit) presented in figure 1.2. Since we have
considered the double cosets {E, σ2 }\C6v /{E, σ0 } (see example 1.2.5) it is convenient to assume that X is
represented by the second configuration, whereas Y by the first one. We can assume these sets to be bases
of a hexagonal prism. From the previous examples (and exercises corresponding to them) one can find the
number of orbits and other properties of G (X ×Y ). For example, there are four orbits, i.e. four nonequivalent
relative positions of two bases of a hexagonal prism with chosen colorings.
1.2.13 Classification of pairs The previous example suggests that Mackey’s Theorem can be applied to pa
classification of oriented pairs (x, y), i.e. elements of a cartesian product X × X provided that X = G(x0 ) is
the orbit of G. Example 1.2.11 together with corollary 1.2.12 yields the following classification scheme:
18
Chapter 1. Classes of configurations
• Each orbit of G on X × X contains an element of the form (x0 , gx0 ).
• The stabilizer G(x0 ,gx0 ) of this pair is given as Gx0 ∩ gGx0 g −1 .
• The set of orbits is in one-to-one correspondence with the set of double cosets Gx0 \G/Gx0 .
• The diagonal ∆(X × X) is an orbit corresponding to the double coset Gx0 1G Gx0 = Gx0 .
• The action of Gx0 on the orbit G(x0 , gx0 ) is similar to the action of Gx0 on the set of left cosets
Gx0 /G(x0 ,gx0 ) and therefore x0 (so also each x ∈ X = G(x0 )) posses exactly |Gx0 /G(x0 ,gx0 ) | neighbors
in the orbit G(x0 , gx0 ) (or of type g, where g is the representative of the double coset Gx0 gGx0 ). In
other words, |Gx0 /G(x0 ,gx0 ) | is equal to the number of pairs in G(x0 , gx0 ) with the first element being
fixed to x0 . In particular, there is only one neighbor of type 1G .
• The pair (gx0 , x0 ) belongs to G(x0 , gx0 ) if and only if g −1 ∈ Gx0 gGx0 . This condition is fulfilled, for
example, by any element g of order 2, i.e. g 6= 1 and g 2 = 1, since in this case we simply have g −1 = g.
In the case of a regular action T on Λ (see 1.1.19) we immediately obtain that all pairs are labelled by t ∈ T
and every point a ∈ Λ has only one neighbor of each type, i.e. such a0 ∈ Λ that a0 − a = t. Note that in this
case pairs (a, t + a) and (t + a, a) belong to different orbits for t 6= 0.
3
To conclude this section we consider the case when we are given actions of two groups on the same set, say
G X and H X. In order to obtain from these an action of G × H on X we have to assume that the given
actions commute with each other, i.e.
for all g ∈ G, h ∈ H, x ∈ X.
g(hx) = h(gx)
Provided this condition is fulfilled, an action
G×H X
is defined by
(G × H) × X → X: ((g, h), x) 7→ g(hx) = h(gx).
commut.act 1.2.14
An easy example is provided by the left and the rigth regular actions of G on G itself. They commute, by
associativity of multiplication in G, and therefore we obtain an action of G × G on G by putting
((g1 , g2 ), g) 7→ g1 (gg2−1 ) = (g1 g)g2−1 .
Exercises
E121
E 1.2.1 Consider a G–set X, a normal subgroup U E G, and the corresponding restriction
the following facts:
U X.
Check
• For each orbit U (x) and any g ∈ G, the set gU (x) is also an orbit of U on X.
• The orbits of U on X form a G/U –set, in a natural way.
• The orbits of G/U on U \\X are just the orbits of G on X.
• The U –orbits which belong to the same G–orbit are of the same order.
E122
E 1.2.2
Prove 1.2.3 and the corollary 1.2.4.
E123
E 1.2.3
Determine all four double cosets from example 1.2.5.
E124
E 1.2.4 Find the transersal of the set of orbits in the example 1.2.7 corresponding to the representatives
determined in exercise 1.2.3. Consider carefully actions of the following subgroups of C6v : {E, C2 , σ0 , σ2 },
{E, C2 , σ1 , σ3 }, {E, C3 , C3−1 σ0 , σ2 , σ4 }, and {E, C3 , C3−1 σ1 , σ3 , σ5 } (the first two are isomorphic with C2v '
D2 , whereas the latter ones with C3v ' D3 ).
E125
E 1.2.5
Prove the statements of 1.2.11.
file:f12.tex
1.3. Classes of configurations
1.3
19
Classes of configurations
Finally we introduce the actions derived from G X and H Y which form our paradigmatic examples, and which
will be discussed in full detail in this and in later sections. In order to prepare this we form the set of all the
mappings from X into Y :
Y X := {f | f : X → Y }.
These mappings are called configurations. If X is finite then its elements may be numbered, say X =
{x1 , . . . , x|X| }, so that each mapping from X to Y can be identified with the sequence of its values:
1.3.1
Y X → Y × Y × . . . × Y : f 7→ (f (x1 ), . . . , f (x|X| )).
{z
}
|
yx
|X| times
Since this mapping is a bijection, then for a finite set Y we obtain the following equation for the number of
mappings (i.e. the number of configurations) in Y X :
|Y X | = |Y ||X| .
1.3.2 Finite crystal lattices In example 1.1.19 we have introduced the translation group and its regular fc
orbit — the crystal lattice Λ. It is easy to see that T can be written as a direct product
T = Z × . . . × Z ≡ Zn
{z
}
|
n times
and, according to 1.3.1, can be identified with the set of mappings t: {1, . . . , n} → Z. Moreover, due to the
bijection between a group and its regular orbit, it is obvious that the nodes of a crystal lattice can also be
labelled by such mappings (or n-tuples (t1 , . . . , tn ), if you wish). Therefore, we can introduce a finite crystal
lattice as a set of mappings from {1, . . . , n} into ZN , where ZN = Z/N Z is the group of residues modulo N ,
or a regular orbit of a finite translation group T = ZnN . Of course, both T and Λ have N n elements in such
case. In this way, we have imposed on a crystal lattice the Born–von Kármán periodic boundary conditions.
There are some several generalizations of these conditions but we will not discuss them here.
3
In order to consider different actions on Y X we start with the most natural one, induced from an action
H Y , which is defined as follows:
1.3.3
H × Y X → Y X : (h, f ) 7→ h̄ ◦ f, h̄ ∈ SY ,
H
i.e. (h, f ) is mapped onto f˜, where f˜(x) := hf (x). The corresponding permutation group on Y X will be
denoted by H̄ E .
The other canonic action can be derived from an action
1.3.4
GX
:
G × Y X → Y X : (g, f ) 7→ f ◦ ḡ −1 , ḡ ∈ SX ,
G
i.e. (g, f ) is mapped onto f˜, where f˜(x) := f (g −1 x). The corresponding permutation group on Y X will be
denoted by E Ḡ .
Combining these two actions, an action of the direct product (H × G) on Y X can be introduced by putting
(see 1.2.14):
1.3.5
(H × G) × Y X → Y X : ((h, g), f ) 7→ h̄ ◦ f ◦ ḡ −1 ,
i.e. ((h, g), f ) is mapped onto f˜, where f˜(x) := hf (g −1 x). The corresponding permutation group on Y X will
be denoted by H̄ Ḡ , and it will be called the power group of H̄ by Ḡ.
Now we are going to introduce the so-called wreath product H oX G and its action on Y X . The previously
defined three actions can be considered as restrictions of this new action to appropriate subgroups. The
H
20
Chapter 1. Classes of configurations
underlying set of a wreath product is the cartesian product of H X (which is a group with respect of pointwise
multiplication: (ψψ 0 )(x) := ψ(x)ψ 0 (x)) and G, i.e.
H oX G := H X × G = {(ψ, g) | ψ: X → H, g ∈ G}.
To determine the multiplication rule we must have an action
GX
at hand, and then
(ψ, g)(ψ 0 , g 0 ) := (ψψg0 , gg 0 ), where (ψψg0 )(x) := ψ(x)ψ(g −1 x).
The actions
expo
GX
and
HY
yield the following natural action of H oX G on Y X :
H oX G × Y X → Y X : ((ψ, g), f ) 7→ f˜, f˜(x) := ψ(x)f (g −1 x).
1.3.6
The corresponding permutation group on Y X will be denoted by [H̄]Ḡ and called the exponentiation group
of H̄ by Ḡ. The orbits of G, H, H × G and H oX G on Y X are called classes of configurations or symmetry
classes of mappings.
A few remarks concerning the wreath product H oX G are in order. They will in particular show that the
actions of G, H, and H × G on Y X are restrictions of the action of H oX G on Y X defined above. The reader
is kindly asked to check carefully the following statements:
1211
1.3.7 Lemma The wreath product H oX G has the following properties:
• The identity element of H oX G is (ι, 1), where ι ∈ H X is the mapping ι(x) := 1, for each x.
• If we define ψ −1 ∈ H X by ψ −1 (x) := ψ(x)−1 (this follows from the pointwise multiplication rule), we
get
−1
−1
(ψ, g)−1 = (ψg−1
), where ψg−1
)g−1 = (ψg−1 )−1 .
−1 , g
−1 := (ψ
• The normal subgroup
H ∗ := {(ψ, 1) | ψ ∈ H X } E H oX G,
is called the base group and it is the direct product of |X| copies H x of H :
H x := {(ψ, 1) | ∀ x0 6= x: ψ(x0 ) = 1H } ' H, for each x ∈ X.
• The subgroup G 0 := {(ι, g) | g ∈ G} ' G is a complement of H ∗ , so that we have
H oX G = H ∗ · G 0 , H ∗ E H oX G, H ∗ ∩ G 0 = {(ι, 1)}.
• The diagonal
∆(H ∗ ) := {(ψ, 1) | ψ constant} ' H,
satisfies
∆(H ∗ ) · G 0 = {(ψ, g) | ψ constant, g ∈ G} ' H × G.
• The wreath product of finite groups H and G contains |H oX G| = |H||X| |G| elements.
This shows that the subgroups G 0 , ∆(H ∗ ) and ∆(H ∗ ) · G 0 are natural embeddings of G, H and H × G into
H oX G, in short:
1217
1.3.8
G ,→ H oX G, H ,→ H oX G, H × G ,→ H oX G,
so that in fact the actions of G, H, and H × G on Y X introduced above are restrictions of the action of
H oX G on Y X . Moreover, since the base group is a direct product of |X| copies of H, we can consider an
action of this group on the cartesian product Y × . . . × Y (cf. 1.3.1) according to 1.2.10
H ∗ × Y X → Y X : ((ψ, 1), f ) 7→ f˜,
f˜(x) = ψ(x)f (x).
1.3. Classes of configurations
21
This action describes, of course, the embedding
H ∗ = H X = H × . . . × H ,→ H oX G.
|
{z
}
|X| times
The corresponding permutation group on Y X will be denoted [H̄]E and it can be called the |X|-fold cartesian
power of H.
The formation of configurations introduced above can be iterated, for example by considering sets of the
form
Z
Y X := Y (W ) ,
where the group acting on this set of mappings is of the form H oZ G. In this way, for example, we can
introduce
1.3.9 Spin models One of the first successes of quantum mechanics was a satisfactory explanation by sp
Heisenberg of the origin of magnetism in crystalline insulators. Magnetic properties of these materials are
determined by some atomic degrees of freedom, localized at nodes of the crystal lattice. Within quantummechanical description, these degrees of freedom are related to internal angular momenta (briefly referred
to as spins) of electrons from partially filled shells in an atom or ion. Spin is determined by the so-called
spin number 1 s = 0, 1/2, 1, . . . and an atom (ion) having spin s can be in one of 2s + 1 states labelled
by m = −s, −s + 1, . . . , s − 1, s, which corresponds to the z coordinate of spin s (spin, related to angular
momentum, is a vector). In the simplest model of magnetic crystals it is assumed that each node of a crystal
lattice carries the same spin s, but spins at different nodes can be in different states m. Therefore, all
possible spin (or magnetic) configurations form the set [−s, +s]X , where [−s, +s] := {−s, −s+1, . . . , s} and
X is the set of crystal nodes. This set is a base for many quantum models of magnetic materials, amongst
which the Ising and the Heisenberg ones are most frequently studied. Note that in this way we have obtained
n
the structure [−s, +s]ZN , i.e. the set of mappings into Y = [−s, +s] from X = ZnN , which itself is a set of
mappings.
3
A detailed discussion of the determination of the symmetry group of a given crystal lattice we postpone to
section 1.6. Here we consider only a nice example which will be examined and generalized later on.
1.3.10 Example It is our aim to define and to enumerate spin configurations on the 3-dimensional cube, hc
i.e. on a 3-dimensional finite crystal lattice with N = 2. So, to begin with, here is the cube, with its vertices
labelled by their coordinates in the 3-dimensional cartesian frame x, y, z:
z
6
001
011
101
111
010
-y
000
100
x
110
Identifying the set {x, y, z} with {0, 1, 2} we can say that a spin configuration on the cube is a mapping
f ∈ Y X , where
X = W Z = {0, 1}{0,1,2}
of crystal nodes into the set Y = [−s, +s]. And a class of spin configurations is an orbit of the symmetry
3
group of the cube on the total set of all these (2s+1)2 spin configurations. Hence, our next step should be
the evaluation of the symmetry group of the cube.
To begin with we note that the addition of 1 modulo 2 in Z2 corresponds to the transposition 0 ↔ 1, so
in this case we can replace the action of Z2 on itself by the action of a symmetric group S2 = {1, τ } (with
τ 0 = 1 and vice versa) on the set W = {0, 1}. Moreover, as we clearly see, permutations of the coordinates
1 This number is connected with the irreducible representations of SU (2), but a precise definition is not necessary at this
stage of our considerations.
22
Chapter 1. Classes of configurations
can be applied. Hence there is also a symmetric group S3 in the game, it acts on the set Z = {0, 1, 2}. Thus
altogether (see 1.3.6) the wreath product S2 o S3 is acting upon X = W Z , and it is a subgroup of the desired
symmetry group. As we know that the symmetry group of the cube, i.e. Oh in the Schoenflies notation, also
has the order 48 = 23 · 3!, we can express it in the following form:
Oh ' S2 o S3 .
Correspondingly, for higher dimensions n, the symmetry group of the n-dimensional cube is the wreath
product
S2 o Sn ,
the so-called hyperoctahedral group of degree n.
3
The result 1.2.12 on double cosets and the action of disjoint sets determined by 1.2.9 suggest some general
remarks on applications, and in particular on applications to classes of configurations.
˙ 2 , where each one of the
Mackdouble 1.3.11 Mackey’s theorem and classes of configurations If X = X1 ∪X
(disjoint) components Xi is assumed to be a union of orbits, then Mackey’s theorem 1.2.12 motivates to
rewrite the set of configurations in the following form:
˙
Y X1 ∪X2 = Y X1 × Y X2 ,
and correspondingly to decompose the two cartesian factors into orbits: If {fij , j = 1, . . . , |G\\Y Xi |} is a
transversal of the orbits of G on Y Xi , for i = 1, 2, then
i [
i h[
h[
˙
˙
˙
˙
[G(f1j ) × G(f2k )] .
G(f2k ) =
Y X1 ∪X2 =
G(f1j ) ×
j
j,k
k
Thus we can rewrite the set of classes of configurations in the following form:
[
˙
˙
G\\Y X1 ∪X2 = G\\Y X1 × Y X2 = G\\ [G(f1j ) × G(f2k )]
j,k
=
[
˙
[G\\G(f1j ) × G(f2k )] .
j,k
Now we are in a position to apply Mackey’s theorem, obtaining the bijection
G\\G(f1j ) × G(f2k ) → Gf1j \G/Gf2k : G(f1j , gf2k ) 7→ Gf1j gGf2k .
This finally yields the desired bijection between the set of classes of configurations and a set of double cosets:
[
G\\Y X −→
Gf1j \G/Gf2k .
j,k
Exercises
E131
E 1.3.1
Check that the action 1.3.4 is well defined, i.e. g1 (g2 f ) = (g1 g2 )f .
E132
E 1.3.2 Show that E Ḡ is normal in H̄ Ḡ and in [H̄]Ḡ , and that H̄ Ḡ is not in general normal in [H̄]Ḡ .
Check that the factor group H̄ Ḡ /E Ḡ is isomorphic to H̄, while [H̄]Ḡ /[H̄]E is isomorphic to Ḡ. What does
this mean, in the light of exercise 1.1.5, for the enumeration of the orbits of H̄ Ḡ and [H̄]Ḡ ?
E133
E 1.3.3 Let X = E ∪ D be set of edges (E) and diagonals (D) of a square. Consider the action of the
symmetry group of the square C4v ' D4 (notice that this group can be also described as a wreath product
S2 o S2 ) on {0, 1}X and find all classes of configurations.
E134
E 1.3.4 Let Y be a set of two colors, say Y = {black, white}. Find all possible colorings of vertices and
edges of a triangle with respect to its symmetry group (which is, of course, C3v ' S3 ).
file:f13.tex
1.4. Finite symmetric groups
1.4
23
Finite symmetric groups
In the first section we mentioned the symmetric group SX on the set X. In order to prepare further examples
and detailed descriptions of actions we need to consider this group in some detail, in particular for finite X.
A first remark shows that it is only the order of X which really matters:
1.4.1 Lemma For any two finite and nonempty sets X and Y , the natural actions of SX on X and SY on sy
Y are isomorphic if and only if |X| = |Y |.
This is very easy to check and therefore left as exercise 1.4.1. We call |X| the degree of SX , of any subgroup
P ≤ SX and of any π ∈ SX . In order to examine permutations of degree n it therefore suffices to consider a
particular set of order n and its symmetric group. For technical reasons we introduce two such sets of order
n:
n := {0, . . . , n − 1} and n := {1, . . . , n},
hoping that it will be always clear from the context if this set n is meant or its cardinality n. It is an old
tradition to prefer the set n and its symmetric group which we should denote by Sn in order to be consistent.
Hence let us fix the notation for the elements of Sn , the corresponding notation for the elements of Sn is
then obvious.
A permutation π ∈ Sn is written down in full detail by putting the images πi in a row under the points
i ∈ n, say
1 ... n
π=
,
π1 . . . πn
what will be abbreviated by
π=
i
.
πi
The points which form the first row need not be written in their natural order, e.g.
123
213
=
.
231
321
We read compositions of mappings from right to left, so that (πρ)(i) = π(ρi). Using the abbreviated notion
of permutations this can be written as follows
i
i
ρi
i
i
πρ =
=
=
.
πi ρi
π(ρi) ρi
(πρ)i
We call a permutation π ∈ Sn a cyclic permutation or a cycle if and only if it can be written in the form
(after changing the order in the first row, if necessary)
i1 i2 . . . ir−1 ir ir+1 . . . in
,
i2 i3 . . .
ir
i1 ir+1 . . . in
where r ≥ 1. In order to emphasize r we also call it an r-cycle, whereas the 2-cycles are called transpositions.
We note that the orbits of the subgroup generated by this particular cyclic permutation are the following
subsets of n: {i1 , . . . , ir }, {ir+1 }, . . . , {in }. We therefore abbreviate this cycle by (i1 , . . . , ir )(ir+1 ) . . . (in ),
where the points which are cyclically permuted are put together in round brackets. For example 11 23 32 =
(2, 3)(1). Commas which seperate the points may be omitted if no confusion can arise (e.g. if n ≤ 10), and
1-cycles can be left out if it is clear which n is meant. Hence we can write π = (i1 . . . ir ) for the r–cycle
introduced above. This cycle π can also be expressed in terms of i1 alone:
π = (i1 πi1 . . . π r−1 i1 ).
Using all these abbreviations and denoting by 1 := (1) . . . (n) the identity element, we obtain for example
S3 = {1, (12), (13), (23), (123), (132)}.
24
Chapter 1. Classes of configurations
The order of a cycle (i1 . . . ir ), i.e. the order of the cyclic group h(i1 . . . ir )i generated by this cycle, is equal
to its length:
|h(i1 . . . ir )i| = r.
1.4.2
Two cycles π and ρ are called disjoint, if the two sets of points which are not fixed by π and ρ are disjoint
sets. Notice that, for example, 1 = (1)(2)(3) and (123) are disjoint cycles. Disjoint cycles π and ρ commute,
i.e. πρ = ρπ, so each permutation of a finite set can be written as a product of pairwise different disjoint
cycles, e.g.
12345678
= (17)(253)(64)(8).
75263418
The disjoint cyclic factors 6= 1 of π ∈ Sn are uniquely determined by π and therefore we call these factors
together with the fixed point cycles of π the cyclic factors of π. Let c(π) denote the number of these cyclic
factors of π (including 1-cycles), let lν be their lengths, ν ∈ c(π) (recall that c(π) = {1, . . . , c(π)}), choose,
for each ν an element jν of the ν-th cyclic factor. Then
sta
1.4.3
π=
Y
(jν πjν . . . π lν −1 jν ).
ν∈c(π)
If we choose the jν so that
134
1.4.4
∀ m ∈ N: jν ≤ π m jν , and ∀ ν < c(π): jν < jν+1 .
then this notation becomes uniquely determined and 1.4.3 is called the (standard) cycle notation for π. We
note in passing that the sets {jν , πjν , . . . , π lν −1 jν } of points which are cyclically permuted by π are just the
orbits of the cyclic group hπi generated by π.
Having described the elements of Sn , we show which of them are in the same conjugacy class, i.e. in the
same orbit of the group Sn on the set Sn under the conjugation action (cf. 1.1.7). In order to do this, we
first note how ρπρ−1 is obtained from the permutation π:
i
i
ρi
i
ρi
ρi
−1
ρπρ =
=
=
.
ρi πi
i
ρ(πi)
i
ρ(πi)
Thus, in terms of cyclic factors of π, ρπρ−1 arises from
π = (. . . i, πi . . .) . . .
by simply applying ρ to the points in the cycles of π:
136
1.4.5
ρπρ−1 = . . . (. . . ρi, ρ(πi) . . .) . . . .
This equation shows that the lengths of the cyclic factors of π are the same as those of ρπρ−1 . It is easy to
see that, conversely, for any two elements π, σ ∈ Sn with the same lengths lν of cyclic factors there exists a
ρ ∈ Sn such that ρπρ−1 = σ. Hence the lengths of the cyclic factors of π characterize its conjugacy class.
To make this more explicit, we introduce the notion of (proper) partition of n ∈ N, by which we mean any
sequence α = (α1 , α2 , . . .) of natural numbers αi which satisfy
X
∀ i: αi ≥ αi+1 , and
αi = n.
i
The αi are called parts of α. The fact that α is a partition of n is abbreviated by
α ` n.
If α ` n then there exists an h such that αi = 0 for all i > h. We may therefore write
α = (α1 , . . . , αh ),
13
1.4. Finite symmetric groups
25
for any such h. The minimal h with this property will be denoted by l(α) and called the length of α. The
following abbreviation is useful in the case when several nonzero parts of α are equal, say ai parts are equal
to i, i ∈ n:
α = (nan , (n − 1)an−1 , . . . , 1a1 ).
If ai = 0, then iai is usually omitted, e.g. (3, 12 ) = (3, 1, 1, 0, . . .). For π ∈ Sn the ordered lengths αi (π),
i ∈ c(π), of the cyclic factors of π in cycle notation form a uniquely determined proper partition
α(π) = (α1 (π), α2 (π), . . . , αc(π) (π)) ` n,
which we call the cycle partition of π. The corresponding n-tuple
a(π) := (a1 (π), . . . , an (π))
consisting of the multiplicities ai (π) of the parts of length i in α(π) is called the cycle type of
Pπ. Correspondingly we call an n-tuple a := (a1 , . . . , an ) a cycle type of n if and only if each ai ∈ N, and
i · ai = n. This
will be abbreviated by
a à n.
The conjugacy class of π ∈ Sn will be denoted by C S (π), the centralizer by CS (π), so that we obtain the
following descriptions and properties of conjugacy classes and centralizers of elements of Sn :
1.4.6 Corollary Let π and σ denote elements of Sn . Then
co
• C S (π) = C S (σ) ⇐⇒ α(π) = α(σ) ⇐⇒ a(π) = a(σ).
• C S (π) = C S (π −1 ), i.e. Sn is ambivalent,, which means that each element is a conjugate of its inverse.
Q
Q
• |CS (π)| = i iai (π) ai (π)!, and |C S (π)| = n!/ i iai (π) ai (π)!.
• |hπi| = lcm {αi (π) | i ∈ c(π)} = lcm {i | ai (π) > 0}.
• Each proper partition α ` n occurs as the cycle partition of some π ∈ Sn .
(The first, second, fourth and fifth item is clear from the foregoing, while the third one follows from the
fact that there are exactly iai ai ! mappings which map a set of ai i-tuples onto this same set up to cyclic
permutations inside each i-tuple.) For the sake of simplicity we can therefore parametrize the conjugacy
classes of elements in Sn (and correspondingly in Sn ) by partitions or cycle types putting
C α := C a := C S (π), when α(π) = α, and a(π) = a.
We note in passing that a1 (π) is equal to the number of fixed points nπ of π (and, hence, of any ρ ∈ C S (π))
acting on n in the natural way.
Since
1.4.7
(i1 . . . ir ) = (i1 i2 )(i2 i3 ) . . . (ir−1 ir )
tr
each cycle, and hence every element of Sn , can be written as a product of transpositions. Thus Sn is
generated by its subset of transpositions (if this is empty, then n ≤ 1, and both S1 = S0 = {1} are generated
by the empty set ∅). But, except for the case when n = 2, we do not need every transposition in order to
generate the symmetric group, since, for 1 ≤ j < k < n, we derive from 1.4.5 that
(j, k + 1) = (k, k + 1)(j, k)(k, k + 1).
Thus the transposition (j, k + 1) can be obtained from (j, k) by conjugation with the transposition (k, k + 1)
of adjacent points. Therefore the subset
Σn := {σi := (i, i + 1) | 1 ≤ i < n} = {(12), (23), . . . , (n − 1, n)},
consisting of the elementary transpositions σi , generates Sn . A further system of generators of Sn is obtained
from
1.4.8
so that we have proved
(1 . . . n)i (12)(1 . . . n)−i = (i + 1, i + 2), 1 ≤ i ≤ n − 2,
tr
26
Chapter 1. Classes of configurations
1.4.9 Corollary
ge
Sn = h(12), (23), . . . , (n − 1, n)i = h(12), (1 . . . n)i.
Another useful result describes the cycle structure of a power of a cycle (the proof is left as exercise 1.4.4):
power
1.4.10 Lemma For each natural number m the power (i1 . . . ir )m consists of exactly gcd(r, m) disjoint
cyclic factors, they all are of length r/ gcd(r, m).
An easy application of 1.4.10 gives
powera
1.4.11 Corollary The elements of order d in the group generated by the cycle (1 . . . n) are the powers
(1 . . . n)i , where i is of the form n·j
d , and 1 ≤ j < d is relatively prime to d.
A direct consequence of this is
subg.cyc
1.4.12 Corollary The group Cm := h(1 . . . m)i contains, for each divisor d of m, exactly one subgroup U
of order d. Furthermore, this subgroup contains φ(d) elements consisting of d–cycles only, if φ(−) denotes
the Euler function
φ(d) := |{i ∈ d | gcd(d, i) = 1}|.
These φ(d) elements form the set of generators of U (i.e. each of them generates U and none of the other
elements does this).
As finite cyclic groups of the same order are isomorphic, they have the same properties:
subg.cyc.a
1.4.13 Corollary A finite cyclic group G has, for each divisor d of its order |G|, exactly one subgroup U
of order d. Furthermore, this subgroup contains φ(d) generators, and so, G has exactly φ(d) elements of this
particular order, which implies
X
φ(d) = n.
d|n
Let us now consider an application of the preceding results to the paradigmatic G–set Y X corresponding to
given Y and G X (cf. 1.3.4).
ex13
1.4.14 Example Let Cp denote the following cyclic subgroup of Sp :
Cp := h(1 . . . p)i ≤ Sp .
It acts on the set X := p = {1, . . . , p} and hence, see 1.3.4, also on the set of configurations Y X := mp ,
which can be considered as the set of all the colorings of the regular p–gon in m colors. For example, in the
case when p = 5 and m = 2, C5 acts on the set 25 , consisting of all the 32 colorings of the regular pentagon
with two colors (black and white), some of which are shown in figure 1.3.
s
Z
s Zs
B
Bs
s
c
Z
s Zs
B
Bc
s
s
Z
c Zs
B
Bs
s
Figure 1.3: Three configurations on the regular 5–gon.
We now assume that p is a prime. Lemma 1.4.10 shows that Cp contains, besides the identity element,
p–cycles only. The identity element of Cp keeps each f ∈ mp fixed, while each p–cycle fixes the m monochromatic colorings only. Hence we obtain from the Cauchy–Frobenius Lemma that
|Cp \\mp | =
1 p
(m + (p − 1)m),
p
Maybe only provided that p is a prime number. In the case when p = 5 and m = 2 we obtain that there exist exactly
a half of this 8 classes of configurations on the regular 5-gon. They are shown in the introduction to the present chapter
picture?
(figure 1.1).
3
1.4. Finite symmetric groups
27
We saw that π = (1 . . . p), p > 2, generates Cp . This permutation, together with
p+1
(1 p−1)(2 p−2) . . . ( p−1
for p odd
2
2 )(p)
1.4.15
σ=
p
(1 p−1)(2 p−2) . . . ( 2 −1 p2 +1)( p2 )(p) for p even
si
forms a generating set for the dihedral group Dp , i.e.
Dp = hπ, σi ,
1.4.16
|Dp | = 2p.
ge
For odd p all elements of the form π k σ, k ∈ p, have the same cycle type (1, p−1
2 , 0, . . . , 0), whereas for even p
there are two classes. For k odd we have the cycle type (0, p2 , 0, . . . , 0) but for k even (2, p2 − 1, 0, . . . , 0). The
dihedral group Dp for p > 2 can be considered as the symmetry group of the regular p–gon, where powers of
π describe rotations and the other elements are reflections. Therefore, we can investigate the action of Dp
on the set of colorings mp defined above.
1.4.17 Example Let us assume that p > 2 is a prime number (and therefore odd). Each of the p reflections ex
fixes colorings monochromatic on appropriate pairs, whereas the only one single point can be in any of m
colors. Then, the Cachy-Frobenius Lemma leads to the following result
|Dp \\mp | =
1
(mp + (p − 1)m + pm(p+1)/2 ).
2p
Substituting, as in the previous example, p = 5 and m = 2 we obtain again 8 orbits. It can be checked by
inspection that each reflection leaves a given configuration in its orbit under C5 . However, it is not a general
case and we will return to this problem at the end of this section.
3
Another important fact exhibits a normal subgroup An of Sn . In order to show this we introduce the sign
(π) as follows:
Y πj − πi
∈ Z, if n ≥ 2, while (1S0 ) := (1S1 ) := 1Z .
(π) :=
j−i
1≤i<j≤n
As i 6= j implies πi 6= πj, we have (π) 6= 0. Moreover, the following orders of sets of pairs are equal:
|{{i, j} | 1 ≤ i < j ≤ n}| = |{{πi, πj} | 1 ≤ i < j ≤ n}|,
and so we have (π) = ±1Z . Furthermore is a homomorphism of Sn into {1, −1}:
Y πρj − πρi Y πρj − πρi Y ρj − ρi
(πρ) =
=
= (π)(ρ).
j−i
ρj − ρi i<j j − i
i<j
i<j
This proves
1.4.18 Corollary The sign map
si
: Sn → {1, −1}: π 7→ (π)
is a homomorphism which is surjective for each n ≥ 2. Hence its kernel
An := ker = {π ∈ Sn | (π) = 1}
is a normal subgroup of Sn :
An E Sn , |An | = |Sn |/2 = n!/2, if n ≥ 2.
The elements of An , the alternating subgroup of Sn , are called even permutations, while the elements of
Sn \ An are called odd permutations. Correspondingly, an r–cycle is even if and only if r is odd. In the case
when G X is a finite action, we can apply the sign map to Ḡ, the permutation group induced by G on X.
Its kernel
Ḡ+ := {ḡ ∈ Ḡ | (ḡ) = 1}
is either Ḡ itself or a subgroup of index 2, as is easy to see. Denoting its inverse image by
G+ := {g ∈ G | (ḡ) = 1},
we obtain a useful interpretation of the alternating sum of fixed point numbers:
28
Chapter 1. Classes of configurations
1.4.19 Lemma For any finite action G X such that G 6= G+ , the number of orbits of G on X which split or
over G+ (i.e. which decompose into more than one — and hence into two — G+ –orbits) is equal to
1 X
1 X
(ḡ)|Xḡ |.
(ḡ)|Xg | =
|G|
|Ḡ|
g∈G
ḡ∈Ḡ
1314
1.4.20 Corollary In the case when G 6= G+ , the number of G–orbits on X which do not split over G+ is
equal to
1 X
(1 − (ḡ))|Xg |.
|G|
g∈G
Note what this means. If G acts on a finite set X in such a way that G 6= G+ , then we can group the orbits
of G on X into a set of orbits which are also G+ –orbits. In figure 1.4 we denote these orbits by the symbol
⊗
..
.
⊗
⊕
..
.
⊕
selfenantiomeric orbits

..
.

enantiomeric pairs
Figure 1.4: Enantiomeric pairs and selfenantiomeric orbits
⊗. The other G–orbits split into two G+ –orbits, we indicate one of them by ⊕, the other one by , and
call the pair {⊕, } an enantiomeric pair of G–orbits. Hence 1.4.19 gives us the number of enantiomeric
pairs of orbits, while 1.4.20 yields the number of selfenantiomeric orbits of G on X. The elements x ∈ X
belonging to selfenantiomeric orbits are called achiral objects, while the others are called chiral. These
notions of enantiomerism and chirality are taken from chemistry, where G is usually the symmetry group of
the molecule while G+ is its subgroup consisting of the proper rotations. We call G X a chiral action if and
only if G 6= G+ . Using this notation we can now rephrase 1.4.19 and 1.4.20 in the following way:
1315
1.4.21 Corollary If
equal to
GX
is a finite chiral action, then the number of selfenantiomeric orbits of G on X is
1 X
(1 − (ḡ))|Xg | = 2|G\\X| − |G+ \\X|,
|G|
g∈G
while the number of enantiomeric pairs of orbits is
1 X
(ḡ)|Xg | = |G+ \\X| − |G\\X|.
|G|
g∈G
The sign of a cyclic permutation is easy to obtain from the equation 1.4.7 and the homomorphism property
of the sign, described in 1.4.18:
alta
1.4.22
(i1 . . . ir ) ∈ An ⇐⇒ r is odd.
But in fact we need not check the lengths of the cyclic factors of π since an easy calculation shows (exercise 1.4.5) that, in terms of the number c(π) of cyclic factors of π, we have
alto
1.4.23
(π) = (−1)n−c(π) , if π ∈ Sn .
Note that the definition of a chiral action based on the parity of (induced) permutation not always corresponds
to the definition based on a decomposition of G into proper and improper rotations. It is easy to check that all
elements of D5 (in general D4k+1 ) are even permutations, so D5 < A5 , though some of them are reflections.
Hence, according to our definition, the action of D5 on m5 is not chiral. However, we can generalize the
1.4. Finite symmetric groups
29
definition of chirality taking into account any subgroup G+ < G with index 2 (it means that G of odd
order acts in an achiral way). Such subgroups are always normal and the homomorphism theorem says that
there exists a natural homomorphism on G with G+ being its kernel. Without loss of generality we can
assume that it is a homomorphism : G → {1, −1} ' G/G+ . Therefore, each subgroup G+ determines its
‘own’ chirality, i.e. with respect to this subgroup and, moreover, 1.4.21 is true (it follows from the assumed
definition of G+ that G 6= G+ ). For example, the cyclic group Cp is always a normal subgroup of the dihedral
group Dp , so it determines ‘cyclic’ chirality of an action of Dp . It is easy to notice that for p = 4k + 3 both
definitions of chirality (permutational and cyclic) lead to the same results.
1.4.24 Example Taking into account formulas obtained in examples 1.4.14 and 1.4.17 together with cy
corollary 1.4.21 we obtain that for a prime number p > 2 there are
m(p+1)/2
selfenantiometric orbits and
1
(mp + (p − 1)m − pm(p+1)/2 )
2p
enantiometric pairs (with respect to the ‘cyclic’ chirality). For example, when p = 5 we have to use at least
m = 3 colors to obtain enantiometric pairs and there are 12 such pairs (representatives of four of them are
presented in figure 1.5, where ‘ ’ represents the third color). It means that 12 orbits of D5 split over C5 . In
fact, substituing p = 5 and m = 3 into appropriate formulas we obtain 51 orbits for C5 and 39 for D5 .
c
Z
Zs
B
Bc
c
c
Z
c Zc
B
B
s
Z
c Zs
B
Bc
s
Z
s Zc
B
Bc
s
Figure 1.5: Representatives of 4 enantiomeric pairs of 5-gon colorings
3
Exercises
E 1.4.1
Prove lemma 1.4.1.
E
E 1.4.2 Show that, for each m, n ∈ N∗ and π ∈ Sn , the permutations π and π m are conjugate, if and only E
if m and each length of a cyclic factor of π are relatively prime.
E 1.4.3 Verify that π, ρ ∈ Sn are conjugates if and only if, for each m ∈ N, c(π m ) = c(ρm ). Hint: solve E
the following system of linear equations:
X
gcd(m, l)(al (π) − al (ρ)) = 0.
l
E 1.4.4
Prove lemma 1.4.10.
pr
E 1.4.5
Check 1.4.23.
E
E 1.4.6
Let D6 act on colorings of a hexagon in m colors. Consider all possible types of chirality.
E
file:f14.tex
30
1.5
Chapter 1. Classes of configurations
Complete monomial groups
We have already met the wreath product H oX G, when G is a group acting on X while H acts on Y . Now
we consider the particular case where G is a permutation group, say G ≤ Sn , and where we take for G X the
natural action of G on n. In this case we shorten the notation by putting
H o G := H on G.
embed
A particular case is H o Sn , the complete monomial group of degree n over H. Many important groups are
of this form, examples will be given in a moment. In the case when H ≤ Sm , then H o G has the following
natural embedding into Smn :
(j − 1)m + i
1.5.1
.
δ: Sm o Sn ,→ Smn : (ψ, π) 7→
(πj − 1)m + ψ(πj)i i∈m,j∈n
j
∗
This can be seen as follows: Recall the direct factors Sm
, for j ∈ n, of the base group Sm
of Sm o Sn (cf. the
j
] acts on the block {(j − 1)m + 1, . . . , jm} as Sm does
remark on H x in 1.3.7). The image of each factor δ[Sm
on m, while the image δ[Sn0 ] of the complement Sn0 of the base group acts on the set of these n subsections
{(j − 1)m + 1, . . . , jm} of length m of the set mn as Sn does act on n. E.g., the element
(ψ, π) := (ψ(1), ψ(2), ψ(3), π) := ((12), (123), 1, (23)) ∈ S3 o S3
is mapped under δ onto
(12)(456) (47)(58)(69) = (12)(475869) ∈ S9 .
| {z } |
{z
}
=δ((ψ,1))
=δ((ι,π))
The image of H o G under δ will be denoted as follows:
H G := δ[H o G]
composition 1.5.2
and it is called the plethysm of G and H. As an application of this permutation representation we obtain a
description of the centralizers of elements in finite symmetric groups. To show this we note that δ[Cm o Sn ],
where Cm := h(1 . . . m)i, is just the centralizer of
σ := (1 . . . m)(m + 1, . . . , 2m) . . . ((n − 1)m + 1, . . . , nm) ∈ Smn .
This follows from δ[Cm o Sn ] ⊆ CS (σ), which is clear from 1.4.5 together with 1.5.1 and the fact that
|CS (σ)| = mn n! = |Cm o Sn | (cf. 1.4.6). The general case is now easy:
142
1.5.3 Corollary If σ ∈ Sn is of type a = (a1 , . . . , an ), then its centralizer CS (σ) is a subgroup of Sn which
is similar to the direct sum
⊕i (Ci Sai ).
Similarly we can show (recall 1.1.7)
143
1.5.4 Corollary The normalizer of the n–fold direct sum
⊕n Sm := Sm ⊕ . . . ⊕ Sm , n summands,
is conjugate to the plethysm Sm Sn .
Thus centralizers of elements and normalizers of specific subgroups of symmetric groups turn out to be direct
sums of complete monomial groups. Since such groups will also occur as acting groups later on, we also
describe their conjugacy classes. Consider an element (ψ, π) in H o Sn and assume that C 1 , C 2 , . . . are the
conjugacy classes of H. If
Y
π=
(jν . . . π lν −1 jν ),
ν∈c(π)
1.5. Complete monomial groups
31
in standard cycle notation, then we associate with its ν-th cyclic factor (jν . . . π lν −1 jν ) the element
144
1.5.5
hν (ψ, π) := ψ(jν )ψ(π −1 jν ) · · · ψ(π −lν +1 jν ) = ψψπ . . . ψπlν −1 (jν )
of H and call it the ν-th cycleproduct of (ψ, π) or the cycleproduct associated to (jν . . . π lν −1 jν ) with respect
to (ψ, π). In this way we obtain a total of c(π) cycleproducts, ak (π) of them arising from the cyclic factors
of π which are of length k. Now let aik (ψ, π) be the number of these cycleproducts which are associated to
a k-cycle of π and which belong to the conjugacy class C i of H (note that we did not say “let aik (ψ, π) be
the number of different cycleproducts”). We put these natural numbers together into the matrix
a(ψ, π) := (aik (ψ, π)).
This matrix has n columns (k is the column index) and as many rows as there are conjugacy classes in H (i
is the row index). Its entries satisfy the following conditions:
1.5.6
aik (ψ, π) ∈ N,
X
aik (ψ, π) = ak (π),
i
X
k · aik (ψ, π) = n.
14
i,k
We call this matrix a(ψ, π) the type of (ψ, π) and we say that (ψ, π) is of type a(ψ, π).
1.5.7 Lemma The conjugacy classes of complete monomial groups H o Sn have the following properties:
• C HoSn (ψ 0 , π 0 ) = C HoSn (ψ, π) if and only if a(ψ 0 , π 0 ) = a(ψ, π).
• The order of the conjugacy class of elements of type (aik ) in H o Sn , H finite, is equal to
Y
|H|n n!/
aik !(k|H|/|C i |)aik .
i,k
• Each matrix (bik ) with n columns and as many rows as H has conjugacy classes, the elements of which
satisfy
X
bik ∈ N,
k · bik = n,
i,k
occurs as the type of an element (ψ, π) ∈ H o Sn .
• If H is a permutation group and α := α(hν (ψ, π)), then the cycle partition α(δ(ψ, π)), where δ denotes
the permutation representation of 1.5.1, is equal to
X
lν · α(hν (ψ, π)),
ν
P
where lν · α, α := α(hν (ψ, π)), is defined to be (lν · α1 , lν · α2 , . . .), and where ν means that the proper
partition has to be formed that consists of all the parts of all the summands lν · α(hν (ψ, π)).
A numerical example is provided by S3 o S2 . The set of proper partitions characterizing the conjugacy classes
of S2 is
{α | α ` 2} = {(12 ), (2)},
the set of corresponding cycle types is
{a | a à 2} = {(2, 0), (0, 1)}.
Thus the types of S3 o S2 turn out to be (we assume that classes of S3 are ordered according to the proper
partitions (13 ), (2, 1), (3), i.e. with the cycle types (3, 0, 0), (1, 1, 0), and (0, 0, 1))

 
 
 
 
 

2 0
0 0
0 0
1 0
1 0
0 0
0 0,2 0,0 0,1 0,0 0,1 0,
0 0
0 0
2 0
0 0
1 0
1 0
co
32
Chapter 1. Classes of configurations

0
0
0
 
1
0
0,0
0
0
 
0
0
1,0
0
0

0
0.
1
The orders of the conjugacy classes are 1, 9, 4, 6, 4, 12, 6, 18, 12. Let us take an element from the fourth
class, e.g. ((12), 1, 1). Under the embedding 1.5.1 in S6 we obtain that its cycle partition is
1 · (2, 1) + 1 · (1, 1, 1) = (2, 1, 1, 1, 1) = (2, 14 ).
Note that in 1.3.10 we have considered the wreath product S2 o S3 and its embedding in S23 = S8 . From
the above considerations it is clear that we have also a natural embedding S2 o S3 ,→ S6 ' S6 different,
of course, from S3 o S2 . However, the embedding considered in this section explains the introduced term
hyperoctahedral group (in 1.3.10 we have rather considered hypercubic group). The wreath product S2 o Sn
(via embedding 1.5.1) acts in a natural way on the set of 2n points in n-dimensional space with coordinates (±1, 0, . . . , 0), (0, ±1, 0, . . . , 0), . . . , (0, . . . , 0, ±1), which are vertices of an n-dimensional analog of the
octahedron. Labelling these points with numbers 1, . . . , 2n we can use the introduced formulas.
We now describe an interesting action of Sm o Sn which is in fact an action of the form
ex14
GY
X
.
1.5.8 Example The action of Sm o Sn on mn is obviously similar to the following action of Sm o Sn on the
set m × n:
Sm o Sn × (m × n) → m × n: ((ψ, π), (i, j)) 7→ (ψ(πj)i, πj).
The corresponding permutation group on m × n will be denoted by
Sn [Sm ]
and called the composition of Sn and Sm , while
G[H]
will be used for the permutation group on Y × X, induced by the natural action of H oX G on Y × X.
The action of the wreath product Sm o Sn on m × n induces a natural action of Sm o Sn on the set
Y X := 2m×n = {(aij ) | aij ∈ {0, 1}, i ∈ m, j ∈ n},
i.e. on the set of 0-1-matrices consisting of m rows and n columns:
Sm o Sn × 2m×n : ((ψ, π), (aij )) 7→ (aψ−1 (j)i,π−1 j ).
Since (ψ, π) = (ψ, 1)(ι, π), we can do this in two steps:
(aij ) 7−→ (ai,π−1 j ) 7−→ (aψ−1 (j)i,π−1 j ).
Hence we can first of all permute the columns of (aij ) in such
the numbers of 1’s in the columns
P a way that P
−1
of the resulting matrix is nonincreasing from left to right:
a
≥
i i,π 1
i ai,π −1 2 ≥ . . . And after having
∗
carried out this permutation with a suitable π, we can find a ψ ∈ Sm
such that the 1’s in each column are now
in succinct positions from top to bottom. This proves that the orbit of (aij ) under Sm o Sn is characterized
by an element of the form


1 ... ... ... 1
.
 ...

..

 ∈ 2m×n ,
1 ... 1

0
i.e. by a proper partition of k := i,j aij . Hence the orbits of Sm o Sn on 2m×n are characterized by the
proper partitions α, where each part αi ≤ n and where the total number of parts is l(α) ≤ m:
P
148
1.5.9 Corollary There exists a natural bijection
Sm o Sn \\2m×n → {α ` k | k ≤ mn, α1 ≤ n, l(α) ≤ m}.
1.5. Complete monomial groups
149
33
Hence an application of the Cauchy-Frobenius Lemma yields the following formula for the number of partitions of this form:
X
−1
1.5.10
2Σν c(hν (ψ,π)) ,
|Sm o Sn \\2m×n | = (m!n n!)
(ψ,π)∈Sm oSn
which can be made more explicit by an application of 1.5.7 since each mapping a : m × n → 2 has to be
constant on cycles of (ψ, π).
3
Exercises
E 1.5.1
Prove that the action δ in 1.5.1 and the following one:
(i − 1)n + j
δ 0 : Sm o Sn ,→ Smn : (ψ, π) 7→
(ψ(πj)i − 1)n + πj i∈m,j∈n
E
are similar. Find δ 0 ((12), (123), 1, (23)) and compare it with the result obtained in this section under δ.
E 1.5.2 Prove that the conjugacy class (in Sn ) of an even element π ∈ Sn splits into two An –classes if and E
only if the lengths of the cyclic factors of π are pairwise different and odd (hint: use 1.5.3).
E 1.5.3
Fill in the details of the proof of 1.5.7.
E
E 1.5.4
Consider S2 o S3 and its embedding in S6 . Give the cycle types of its conjugacy classes.
E
E 1.5.5 Note that S2 oS2 can be embedded in S4 considering its action on the set 22 (as the hyperoctahedral E
group with the following bijection 22 → 4: (0, 0) 7→ 1, (1, 1) 7→ 2, (0, 1) 7→ 3, (1, 0) 7→ 4) and on the set 2 · 2
(as has been done in this section). Compare these two embeddings and show that S2 o S2 ' D4 .
file:f15.tex
34
1.6
Chapter 1. Classes of configurations
Hidden symmetries and fibrations
In the previous sections we have presented groups which are important for further considerations: cyclic,
dihedral, alternating, symmetric and complete monomial groups. However, we still have not discussed the
most important for our applications, the symmetry group of a crystal lattice. We are not going to give an
extensive description of this problem, since there are many books devoted to this problems, so there is no
need to repeat their results here. The main aim of this section is to derive the symmetry group of a finite
hypercubic lattice, which is necessary for investigations of spin configurations determined on this lattice.
Recall that a crystal lattice (finite or infinite) has been defined as a regular orbit of the translation group T
(see 1.1.19 and 1.3.2). It is the most trivial case of the symmetry of a set X induced by a symmetry group
G. In the general case we are given an action G X and the symmetry itself is, in a mathematical sense, the
decomposition G\\X of X into its orbits under G. But there may also be relationships between different
orbits, for example, some of them may have the same order, and so on. A certain kind of such further
relationships can be described in a natural way by a bigger, but not too big, a group that also acts on X, as
we are going to describe now.
The action of G on X gives the same orbits as the action of the image
Ḡ = {ḡ | ḡ: x 7→ gx, g ∈ G} ≤ SX
of G in the symmetric group SX . Somewhere between Ḡ and the symmetric group SX there is the normalizer
of Ḡ (cf. example 1.1.7):
NSX (Ḡ) := {π ∈ SX | π Ḡ = Ḡπ}.
Since this group is a subgroup of the symmetric group, it consists of permutations of X, and so we have a
natural action of the normalizer on X. Here is an example of a normalizer:
leftcoset
1.6.1 Example We consider, in S6 , the subgroup of order 2, generated by the permutation
(12)(34)(56),
obviously an embedding S¯2 of S2 . Its normalizer, according to corollary 1.5.3, is
NS6 (S¯2 ) = C2 S3 = S2 S3 .
3
It is easy to note that elements of the obtained normalizer permute members of each coset separetely or/and
permute whole cosets. It can be shown (exercise 1.6.1) the same result can be obtained for any subgroup
U ≤ G, i.e. NSG (U ) = SU SU \G . This fact is generalized by the following
1.6.2 Lemma The orbits of NSX (Ḡ) on X are unions of orbits of G. Moreover, the orbits of G on X are
blocks of NSX (Ḡ), i.e. each ω ∈ G\\X is mapped under any π ∈ NSX (Ḡ) either on itself or on a subset of X
that is disjoint to ω. In particular, NSX (Ḡ) permutes the orbits of G on X.
Thus there exists a natural symmetry between the orbits of G which is induced by its normalizer in SX . We
call this symmetry the hidden symmetry of G on X, while the orbits of G on X exhibit the obvious symmetry.
H. Weyl pointed to this fact at the end of his famous booklet on symmetry, and therefore we should like to
call Weyl’s Recipe the recommendation, to consider both the obvious and the hidden symmetries.
hidden
1.6.3 The holomorph of a group A very nice hidden symmetry exists if the action G X is regular, as we
have in the case of crystal lattices. According to 1.1.18 such an action is similar to the action of G on itself
via left multiplication. Hence we may replace X by G, and so, in this particular case, the hidden symmetry
is induced by the normalizer of Ḡ ' G in the symmetric group SG . This group is called the holomorph of G:
hol (G) := NSG (Ḡ).
h1
1.6.4 Lemma The stabilizer (hol G)1 of the identity element is the automorphism group Aut G of G.
1.6. Hidden symmetries and fibrations
35
Proof: To begin with we show that (hol G)1 is contained in Aut G. The holomorph hol G is a subgroup of
SG , and the ḡ are also elements of SG . Hence, for each h ∈ hol G, the mapping ḡ 7→ hḡh−1 is a conjugation in
this symmetric group, and therefore it is an inner automorphism of SG . Since Ḡ is normal in the holomorph,
this mapping is also an automorphism of Ḡ.
Such a mapping, being an automorphism, is also a permutation of Ḡ. The mapping g 7→ ḡ is an isomorphism,
and so the inverse image of this permutation, which is the mapping g 7→ hgh−1 is also a permutation, but a
permutation of G. These permutations are related by the identity
hḡh−1 = hgh−1 .
This last permutation on G is the same as the one introduced by h: Using the last identity we obtain
hgh−1 h(1) = hḡh−1 h(1) = hḡ(1) = h(g).
On the other hand, since h(1) = 1, we also get
hgh−1 h(1) = hgh−1 = hgh−1 .
This proves that (hol G)1 is contained in the automorphism group Aut G of G.
If, conversely, α denotes an automorphism of G, then α fixes the identity element, and it is a permutation
of G. Hence Aut G is contained in (hol G)1 , which completes the proof.
2
Hence the holomorph contains both the subgroup Ḡ and the automorphism group Aut G. Moreover, these
subgroups have clearly a trivial intersection:
Ḡ ∩ Aut G = {1}.
Since the order of the holomorph is the product of the orders of these groups (recall 1.1.10), it must in fact
be the product of them:
1.6.5
hol G = Ḡ · Aut G
ho
and this means that it is a semidirect product, i.e.
1.6.6
hol G = Ḡ ×id Aut G = Ḡ o Aut G.
ho
It is easy to define an action of hol G on a regular orbit X of G (recall that elements of X are labelled as
xg = gx, where x is an arbitrarily chosen represenatative of X = G(x) and Ḡ < SG )
1.6.7
(Ḡ o Aut G) × G(x) → G(x): ((ḡ, α), g 0 x) 7→ (ḡα(g 0 ))x.
se
An interesting example is the holomorph of a cyclic group Cm . Assuming that x generates Cm , we note that
each automorphism of Cm is generated by an exponentiation x 7→ xi , and it is clear that such a mapping
is an automorphism if and only if i and m are relatively prime: gcd(i, m) = 1. Hence there are exactly
φ(m) different automorphisms, and so |Aut Cm | = φ(m). In the case when m = p, an odd prime, then
φ(m) = φ(p) = p − 1, and so |hol Cp | = p(p − 1), an even number. It is not difficult to see, that this
holomorph contains the dihedral group Dp .
3
Now we are in the position to discuss the symmetry group of a given infinite crystal lattice Λ. Since this
lattice is a regular orbit of T ' Zn , then we know the obvious symmetry, which in this case even defines
X = Λ. According to the presented scheme we are going to derive the corresponding hidden symmetry. The
automorphism group of T consists of the invertible linear transformations over Z, i.e.
Aut T = GL(n, Z).
36
Chapter 1. Classes of configurations
This is clear from the fact that each automorphism corresponds to a new choice of generators b0i , i.e.
b0i =
1.6.8
X
aki bk ,
k∈n
where (aki ) is an invertible Z-matrix, what means that it must have det(aki ) = ±1. The hidden symmetry
has here an obvious crystallographic meaning: A choice of generators is equal to a choice of an elementary
Bravais cell together with the convention determining the order of the generators (i.e. the Bravais cell is
defined here by a sequence rather than by a set of generators).
Since for each pair of basis B and B0 of Rn there exists an automorphism in GL(n, R) then all crystal lattices
form an orbit with a lattice stabilizer beeing GL(n, Z). In this sense all lattices, as members of the same orbit,
are equivalent. However, we have to take into account an orthogonal form the space Rn is equipped with
(cf. the remark in the definition of a crystal lattice 1.1.19). Therefore, we decompose this unique orbit into
orbits of the group O(n, R) < GL(n, R) which consits of automorphisms preserving a given orthogonal form.
Hence, we obtain the bijection between crystal lattices and the double cosets O(n, R)\GL(n, R)/GL(n, Z)
and a crystal lattice is fixed under a cross-section P = O(n, R) ∩ GL(n, Z). Depending on a chosen basis B,
i.e. also on a lattice Λ(B), these stabilizers can be different but they are always finite. Since for each p ∈ P
and 0 = (0, . . . , 0) being the identity element in T we have p0 = 0 then the same holds for an arbitrarily
chosen the origin of a lattice Λ (this point is labelled by 0 ∈ T ) and therefore P is called a point group. A
semidirect product (contained in a holomorph hol T ) T o P is called a space group.
To stop this discussion here we only mention some very important features. Up to now we have been
considering only a purly mathematical description of crystal lattices consisting of “bare” points in the
Euclidean space over R. In the physical reality these points may carry groups of atoms (e.g. molecules
as NaCl or H2 O). A lattice is still invariant under translations, provided that at each node a molecule is
oriented in the same way with respect to a given external reference frame, but some point group operations
are forbidden. Moreover, some atoms
P (or ions) may occupy positions which are described by translations
with rational coefficients (i.e. by t = i qi bi with qi ∈ Q). These facts lead to the notion of group extensions
and only within this framework complete description and classification of space groups can be done. The
introduced above space groups, being semidirect products, form a class of the so-called symmorphic space
groups (amongst as many as 230 space groups in R3 only 73 are symmorphic ones!). Furthermore, they are
maximal point groups, what means that no point group operations are forbidden. However, we will remain
with these simplifications in order to obtain as clear as possible description of spin configurations, so we
assume that spins are carried by crystal nodes only (in this way we remove even ions or atoms from our
game) and no restrictions are imposed on elements of a point group. Moreover, we limit our considerations
to the simple hypercubic lattices which are presented and investigated in the following example.
hypercube
1.6.9 Hypercubic lattices Let E = {e1 , e2 , . . . , en } be a given orthonormal basis in the Euclidean space
Rn , i.e. (ek , el ) = δkl , where (−, −) denotes a scalar product in Rn . The n–dimensional hypercubic lattice
is determined by the basis B = {b1 , b2 , . . . , bn } consisting of n vectors of the same length b = |bk |, for all
k ∈ n, which are parallel to the vectors of the basis E, i.e. bk = bek and therefore (bk , bl ) = b2 δkl . We are
looking for n × n matrices over Z, which preserve the scalar product, i.e. such that (b0i , b0j ) = b2 δij , where b0i
is determined by 1.6.8. This condition yields the following system of equations
X
a2ki = 1
k∈n
X
aki akj
=
0
for i 6= j.
k∈n
The only possible integer solutions are provided by matrices which contain in each row and each column
only one element not equal to zero and, moreover, this element can be only ±1. Let πi, where π ∈ Sn , be a
row-index of this unique nonzero element in the i-th column. Hence, all elements can be written as
aki = ψ(k)δk,πi ,
where ψ is a mapping ψ: n → {−1, +1} < Z. Recall that if transformation of bases is described by a matrix
a then coordinates of vectors are transformed by its inverse. In the case of orthogonal matrices the inversion
ba
1.6. Hidden symmetries and fibrations
37
means the same as transpostion, so vectors t = (t1 , t2 , . . . , tn ) are transformed by the matrix A = a−1 with
matrix elements
Aik = aki = ψ(i)δi,πk = ψ(i)δπ−1 i,k
and we obtain
(At)i =
X
k
Aik tk =
X
ψ(i)δπ−1 i,k tk = ψ(i)tπ−1 i .
k
It is easy to notice that a pair (ψ, π) is an element of the complete monomial group C2 o Sn , i.e. the
hyperoctahedral group, which acts in a natural way on a set consiting of the basis B and its inverse counterpart, i.e. −B := {−b1 , −b2 , . . . , −bn }. Note that the group C2 = {1, −1} is the automorphism group
of Z, which acts simply by the multiplication. In this way we have found the hidden symmetry of the
considered lattice, which moreover preserves the given orthogonal form, and the whole space group is given
as a semidirect product
1.6.10
Zn o (C2 o Sn )
hy
and its action on T , and hence on Λ, is given by the following formula (see 1.6.7)
1.6.11
(t, (ψ, π))t0 = t + t̃0 where t̃0 (i) = (ψt0π )(i) = ψ(i)t0 (π −1 i),
hc
where elements of T are presented as mappings t: n → Z. The obtained group can be written in a more
compact form as
1.6.12
hy
hol Z o Sn ,
where hol Z = Z o C2 (recall that C2 ' Aut Z). An isomorphism η : Zn o (C2 o Sn ) → hol Z o Sn is defined as
η(t, (ψ, π)) = ((t, ψ), π).
To prove this it is enough to check a homomorphism property and for any elements (t, (ψ, π)) and (t0 , (ψ 0 , π 0 )) R
we have
th
M
η[(t, (ψ, π))(t0 , (ψ 0 , π 0 ))] = η[(t + ψt0π , (ψψπ0 , ππ 0 ))]
ex
= ((t + ψt0π , ψψπ0 ), ππ 0 ) = ((t, ψ)(t0π , ψπ0 ), ππ 0 )
= ((t, ψ)(t0 , ψ 0 )π , ππ 0 ) = ((t, ψ), π)((t0 , ψ 0 ), π 0 ).
To end this long story we only mention that many, but not all, point groups are subgroups of the hyperoctahedral group, so their action on Zn can be considered as restriction of this one presented above.
3
In this book we are within the framework of the so-called finite lattice method, what means that we are
investigating properties of models defined on a finite lattice (with the Born–von Kármán periodic boundary
conditions) to extrapolate obtained results to the case N → ∞. Therefore, it is necessary to consider hidden
symmetries of finite lattices. However, there are some problems. At first, the automorphism group of
ZN ' CN has very rich structure, which depends on the arithmetic structure of N . Moreover, realtions
between Aut ZN and Aut ZnN are quite complicated (|Aut Z2 | = 1, |Aut Z22 | = 6, |Aut Z32 | = 168) and they give
rise to number theoretic considerations. Secondly, we want a finite lattice to “mimic”an inifte one, first of
all the orthogonality properties induced by the scalar product in Rn . However, ZN in general is a ring not
a field, so we can not introduce a scalar product in ZnN . These problems are solved by an epimorphism of
actions introduced in section 1.1. This means we have to define a mapping θ: Zn → ZnN and an epimorphism
η: P → P 0 ≤ Aut ZnN such that for all t ∈ T and p ∈ P we have η(p)θ(t) = θ(pt). (Since in these considerations
we have two different translation groups then, to avoid ambiguity, elements of a finite one will be denoted
by τ .) It is easy to define θ as a ring epimorphism, i.e. (θt)(i) = t(i) mod N . This epimorphism allows us to
define η. Let T = {tτ | τ ∈ ZnN } be a transversal of the cosets determined by ker θ, i.e. θ(tτ ) = τ . Then η can
be defined as follows
η(p)τ = θ(ptτ ).
38
Chapter 1. Classes of configurations
We need to check four conditions: (i) η is a homomorphism from P into Aut ZnN ; (ii) η(p) is an automorphism
for each p; (iii) the pair (η, θ) determines an epimorphism of actions P on ZN and P 0 = im η on ZnN ; (iv)
this epimorphism does not depend on a choice of the transversal T .
Hereafter we will refer to P 0 = im η as a modular image of a point group P . The introduced epimorphisms
determine also a modular image of a (symmorphic) space group as
(η, θ): T o P → θT o ηP = ZnN o P 0 .
It is not difficult to check that appropriate actions are epimorphic.
fhypercube
1.6.13 Finite hypercubic lattices Let us consider a modular image of the the hyperoctehedral group
and next the corresponding space group. Without loss of generality we can choose a transversal in T / ker θ
to be tτ = τ (it is done in a formal way since t and τ are elements of different groups, but τ (i) for each i is
an integer). Each p ∈ P can be written as a pair (ψ, π), so we obtain
η(p)τ = θ((ψ, π)τ ) = θ(ψ(1)τ (π −1 1), . . . , ψ(n)τ (π −1 n)).
Since for any integer 0 ≤ z < N we have
−z mod N = N − z = (N − 1)z mod N
then
η(p) = (η(ψ), π) where η(ψ)(i) =
1
N −1
for ψ(i) = 1,
for ψ(i) = −1.
Because for any N there is gcd(N, N − 1) = 1, then multiplication by N − 1 is an automorphism of ZN . In
this way have obtained that
C2 o Sn for N > 2,
η(C2 o Sn ) =
Sn
for N = 2.
The C2 group in the right-hand side is generated by the automorphism z 7→ N − z for z ∈ ZN . Since
Aut Z2 = {id } then we must have η(ψ)(i) = 1 for all i ∈ n. An application to the space group is trivial and
the result can be written in the similar form as 1.6.12, i.e. the space group of finite hypercubic lattice is
DN o Sn for N > 2
and Z2 o Sn for N = 2,
where we have used the fact that Zn o C2 ' DN (cf. 1.4.15 and 1.4.16). This result shows why in example
1.3.10 we obtained the hyperoctahedral group S2 o S3 as the space group of the lattice with the Born–von
Kármán period N = 2.
3
We have seen in example 1.6.1 that the obvious symmetry determined by the regular action of a given
subgroup U on a group G yields the hidden symmetry described by a plethysm SU SG/U . Below we show
general construction, the so-called fibration, which leads to the action of group plethysms or compositions.
Moreover, we show how for given actions G X and H Y new actions induced by them, their direct product,
composition and exponentiation, can be formulated within uniform scheme. We begin with consideration of
group fibration to illustrate basic concept.
groupfib
1.6.14 Fibration of a group Let G be a group, U ⊆ G one of its subgroups, and T any transversal of
right cosets U \G. The corresponding decomposition
[
G=
Ut
t∈T
of the group G into right cosets, together with a choice of representatives, provides a unique presentation of
the elements g ∈ G in the following form
g = ut,
1.6. Hidden symmetries and fibrations
39
with t ∈ T , and where u ∈ U is a uniquely determined element of U . Thus, the element t specifies the coset
of g, while u specifies the position of g relative to t in this coset. In other words, we have a mapping
µ: G → U × U \G: g 7→ (u, U t),
which imposes the structure of the cartesian product U × U \G on the set G. But this structure is not
canonical, i.e. it does not result solely from the definitions of G, U \G and U , since it involves a choice of the
representatives t ∈ T . We are free to make any other choice, t0 = c(U t)t, by means of an arbitrary mapping
c: U \G → U , often referred to as gauge transformation. It yields another mapping µ0 , and thus another
structure of the cartesian product on G. Putting µ0 (g) = (u0 , U t0 ), we obtain
U t0 = U t,
and
u0 = uc(U t)−1 .
Thus, the gauge transformation c does not affect the coordinate index U t of g, but changes its U -coordinate.
Loosely speaking, U t is “absolute”, whereas u is “relative”. The mapping
ν: G → U \G: g 7→ U g
is called a fiber bundle, the cosets U g are called the fibers, and the choice of representatives is called a section.
3
This suggests the following generalization of the notion of cartesian product of sets. Let E, X, Y be sets,
and assume p: E → X to be a surjection with fibers of the same order as a reference set Y :
∀ x ∈ X: |p−1 (x)| = |Y |.
Then
ξ = (E, X, Y, p)
is called a fibration of the set E by p. The sets E, X, Y , and p−1 (x) are then called a bundle, a base, a
typical fiber,
S and fiber over x ∈ X, respectively. The mapping p: E → X is called a bundle projection and
ν : E → ˙ x∈X p−1 (x) is a fiber bundle. (See also figure 1.6.)
1.6.15 Example Let us consider a cartesian product E = X × Y . By definition, E is already equipped
with two canonical surjections onto each of its factors:
p1 : E → X: (x, y) 7→ x,
p2 : E → Y : (x, y) 7→ y.
By choosing, for example, the first mapping, we obtain the fibration ξ = (E, X, Y, p1 ) and the fiber over
x ∈ X is
p−1
1 (x) = {(x, y) | y ∈ Y }.
3
This example demonstrates that the fibration can be looked at as a generalization of cartesian product. It
involves only the projection p1 onto the first factor, and it forgets about the projection p2 , requiring instead
only that cardinalities of all images p−1
1 (x), x ∈ X should be the same. Thus, the fibration ξ = (E, X, Y, p1 )
admits various structures of cartesian product X × Y . These structures differ mutually by permutations
within each fiber p−1
1 (x) ⊆ E, x ∈ X.
1.6.16 Group-subgroup relation The decomposition of a group G into right cosets with respect to its
subgroup U shown above yields the fibration
ξ = (G, U \G, U, ω).
Here G is a bundle, the set of cosets is a base, the subgroup U itself serves as the typical fiber, and the
bundle projection is the mapping
ω: G → U \G: g 7→ U g.
The corresponding coset is the fiber over U t, t ∈ T i.e.
ω −1 (U t) = U t.
(Please do not be confused. On the left-hand side U t is an element of the set of cosets, but on the righthand
side it is this coset itself, i.e. a set of elements of G of the form ut with u ∈ U .)
3
40
Chapter 1. Classes of configurations
It is natural to give the following definition for the equivalence of fibrations. Two fibrations, ξ = (E, X, Y, p),
and ξ 0 = (E 0 , X 0 , Y 0 , p0 ), are equivalent if and only if there exist bijections α: E → E 0 and β: X → X 0 such
that the following diagram
α
E −→
E0
p↓
↓ p0
β
X −→ X 0
is commutative. For example, let U 0 = g0 U g0−1 be a conjugate of U by an inner automorphism of G. Then the
fibrations ξ = (G, U \G, U, ω) and ξ 0 = (G, U 0 \G, U 0 , ω 0 ) are equivalent. The equivalence can be estabilished
by bijections α: G → G: g 7→ g0 gg0−1 and β: U \G → U 0 \G: U t 7→ U 0 t0 , where U 0 t0 = g0 U tg0−1 = U 0 (g0 tg0−1 )
(see exercise 1.6.2).
If this exercise is too trivial you may remove both, this \ref{E1611} and the exercise itself. WSF
Having defined actions G X and H Y one can determine different actions on a given fibration ξ(E, X, Y, p).
These actions will be consitent with the fibration structure if they yield a commutative diagram
E
p↓
X
γ
−→
E
↓p
X
g
−→
where γ denotes an action on the fibration induced by
GX
or/and
HY
.
1.6.17 Example The structure of the cartesian product X × Y is described by two canonical projections p1
and p2 onto each factor. Therefore, we should require that for both fibrations ξ(E, X, Y, p1 ) and ξ(E, Y, X, p2 )
the obtained diagrams are commutative. It is easy to check that this condition is satisfied by the action of
a cartesian product 1.2.10 of G × H on X × Y , i.e. by
((g, h), (x, y)) 7→ (gx, hy).
3
Considering a fibration ξ(E, X, Y, p) we have much more freedom, since it involves only one canonical projection. Therefore the action induced by G X remains essentially the same as in the case of cartesian product,
but that of H on the typical fiber allows independent permutations of elements in each fiber p−1 (x), x ∈ X.
Thus, a general form of permutation imposed by the fiber action H Y is given by an arbitrary mapping
ψ: X → H, such that for e ∈ p−1 (x) we have
e 7→ ψ(x)e ∈ p−1 (x).
The set of all such mappings forms the group H X , a base group of the wreath product H oX G. To consider
an action of H oX G on ξ(E, X, Y, p) we introduce the so-called cartesian map on a bundle.
Any bijection Ψx : p−1 (x) → Y provides a local coordinate system for the fiber p−1 (x). Clearly, it implies
−1
existence of the inverse bijection Ψ−1
(x), which produces a faithful copy Yx of the typical fiber Y
x :Y → p
over a base element x. However, none of these bijections is canonical, in contrary to the projection p2 for a
cartesian product. A complete set {Ψx | x ∈ X} of such bijections defines a cartesian map on a bundle E,
such that
e 7→ (x, y), where x = p(e), y = Ψx (e).
Using this map, we can specify the action of the wreath product H oX G on the bundle set E as
(x, y) 7→ (gx, ψ(gx)y)
for (ψ; g) ∈ H oX G.
Comparing this formula with that given in example 1.5.8 we can see that except for irrelevant change of
components order it is essentialy the same formula. It means that in fact we considered the action on a
fibration not on a direct product. It is clear since the sets m and n have not “equal rights”— the first is a
set of elements in a block, whreas the second is a set of blocks (cf. also exercise 1.5.4).
1.6. Hidden symmetries and fibrations
41
Considering the fibration of a group we have called the choice of representatives a section of the bundle.
Formal definition says: For a given fibration ξ = (E, X, Y, p) any mapping φ: X → E which satisfies
p ◦ φ = id X ,
is called a section of the bundle E. The set
s(E) := {φ: X → E | p ◦ φ = id X }
of all sections of the bundle E is in a one-to-one correspondence with the set Y X of all configurations due
to the following bijection
s(E) → Y X : φ 7→ f : f (x) = Ψx (φ(x)), ∀ x ∈ X,
and for a given cartesian map {Ψx | x ∈ X} of the bundle E. Therefore, it is possible to determine the
action of H oX G on s(E) in a standard way. Note, that H oX G acts as a composition on a bundle E, whereas
it acts as an exponentiation on s(E). The first set has order |X| |Y |, but the second one — |Y ||X| .
The structures presented in all these examples are illustrated by the following picture:
Ψx
section
s
c
c
s
I c@ c
@
s @
c @c
YH
sH
c
c
Y
H
H c
s
c
s c
c
typical fiber
?
?
base
?
p over x
fiber
?
x
c
c
c
c
c c
c
c
c
c
c
c
x
Figure 1.6: Fibration and related structurs
Exercises
E 1.6.1 Let U ≤ G acts on G via right (or left) regular action. Show that in this case NSG (U ) = SU SG/U E
(SU SU \G , respectively).
E 1.6.2 Show that for any transversal T of left (or right) cosets G/U (U \G, respectively) T 0 = gT g −1 is E
a transversal of corresponding cosets with respect to U 0 = gU g −1 .
E 1.6.3
Check that the action in 1.6.7 is well defined.
file:f16.tex
E
42
1.7
Chapter 1. Classes of configurations
Graphs and interactions
We have introduced a crystal lattice as a set of mappings and we have found its symmetry group taking into
account Weyl’s Recipe and the orthogonality condition. We have also mentioned the so-called spin models,
the underlying set of which consists of all spin configurations, i.e. of all the mappings Λ → [−s, +s] (recall
that Λ itself can be described as a set of mappings n → ZN ). Now we are going to consider interactions
between objects, which are determined by a Hamilton function, in classical physics, or by a Hamilton operator
(Hamiltonian, for short), in quantum description. This can also be described by group actions, but to do
this we need some additional definitions.
At first we note that a given action G X determines an action of G on the so-called power set of X, i.e. on
2X := {M | M ⊆ X}, in the following way:
G × 2X → 2X : (g, M ) 7→ gM := {gm | m ∈ M }.
ksets
(This notation uses the fact that an f ∈ 2X can be identified with the subset M ⊆ X where f takes the value
1. Or, in other words, f can be considered as the characteristic function χ: X → {0, 1} of that subset.) It is
not difficult to see that the orbit G(M ) consists of subsets of the same cardinality |M | = k. This suggests
to consider the subaction of G on the set
X
1.7.1
:= {M ⊆ X | |M | = k}
k
of k–subsets (for fixed k = 1, 2, . . . , |X|), obtaining then an action of G on the set of mappings ϕ:
too.
int.obj
X
2
→ Y,
1.7.2 Interacting objects Any potential-energy function (or an opertor in quantum physics) Φ describing
interactions among n = |X| identical objects (particles) can be quite generally resolved into one-body, twobody etc. contributions as follows:
n
X
X
Φk (M ),
Φ(X) =
k=0 M ∈(X )
k
where each Φk describes interactions of exactly k objects. The first term, corresponding to M = ∅, is
constant, hence it can be omitted. One-particle terms describe, as a rule, external forces (fields), but the
so-called on-site interactions can also be introduced. It is clear from this formula that we have assumed
symmetric potentials, i.e. they do not depend on orientations in subsets M . The above representation can
be useful in the usual types of theoretical modeling if the energy of the k-particle interactions tends to zero
with increasing k. Nevertheless, in many cases we have to include the third term or even the fourth one.
The potentials Φk (M ) may have quite complicated structure depending on the properties of objects we want
to take into account and on other model assumptions. One of the simplest example is provided by the spin
interactions in the Ising model. At each lattice node i ∈ Λ there is a spin σi = ±1 (so σ is a mapping
Λ → {−1, +1}). The pairwise interactions between spins are determined by their product σi σj and the
so-called exchange integral Jij ≡ J({i, j}) leading to the term Jij (σi σj ). The last factor (distinguished by
the parentheses) depends on a pair {i, j} only in an indirect way, i.e. via the spin configuration σ restricted
to {i, j}. In general, this term is constructed from operators which
“measure” a given variable (like electric
charge, spin projection, mass and so on) at points i ∈ M ∈ X
.
This
term of a given Hamiltonian is usually
k
most complicated and our example is indeed very simple. Moreover, it is this term which actually acts
(operates) on configurations. The first term, Jij in the considered
case, does depend on M and, moreover,
only on M . In the general case this term is a mapping J: X
→
R
and hereafter it will be referred to as
k
a configuration of exchange integrals (due to applications in spin models). It is assumed very often that
possible values of J form a finite set, e.g. {±1, 0}. It should be underlined that J describes the strength
of the interaction between points in M , idependent of their properties, whereas the form of the operator
considered above determines a type of interaction. For a given type of interactions we can consider different
models depending on the configuration of exchange integrals.
3
1.7. Graphs and interactions
43
The most important interactions, according to the previous example, are pairwise ones, i.e. those between
two objects. Therefore, they quite generally can be described by mappings determined on X
2 . To consider
these problems we introduce the notion of (unlabelled) graphs which can very well serve as a quite general
interaction model since pairwise interactions are indicated by edges.
1.7.3 Graphs The way of defining graphs as orbits of groups may at first glance seem to be circumstantial, ex
but we shall see that this definition is very flexible since it can easily be generalized to all other kinds of
graphs like multigraphs, directed graphs and so on
• A labelled (simple) graph consists of a set of vertices and a set of edges joining pairs of vertices, but
neither loops (i.e. edges joining a vertex with itself) nor multiple edges are allowed. Thus a labelled
graph on v vertices can be considered
(after numbering the vertices from 1 to v, say) as a map f from
the set of pairs of vertices v2 into the set Y := 2 = {0, 1}, where we put
f ({i, j}) :=
n
1
0
if an edge joins i and j,
otherwise.
For example, the first one of the two labelled graphs of figure 1.7 can be identified in this way with
the mapping f : 24 → 2 defined by
{1, 2} 7→ 0, {2, 3} 7→ 1,
f : {1, 3} 7→ 0, {2, 4} 7→ 0,
{1, 4} 7→ 1, {3, 4} →
7 1.
v
• The symmetric group Sv acts on v and hence also on v2 , so that we obtain an action of Sv on 2(2)
which is of the form G (Y X ) as was described in 1.3.4. Two labelled graphs are called isomorphic if
and only if they lie in the same orbit under this action, i.e. if and only if each arises from the other by
renumbering the vertices, so that for example the labelled graphs of figure 1.7 are isomorphic (apply
π := (34) ∈ S4 ).
• A graph Γ on v vertices is defined to be such an isomorphism class of labelled graphs. It can be
visualized by taking any member of the isomorphism class and deleting the labels. This yields for the
labelled graphs shown in figure 1.7 the drawings of figure 1.8. It should be clear by now what we mean
by a graph, and that in our terminology a graph is not a pair (V, E) consisting of a set V of vertices
and a set E of edges, but that a graph Γ can be represented by such a pair, so that, for example, the
graphs of figure 1.8 are in fact equal.
• If we want to allow multiple edges, say up to the multiplicity k, then we again consider X := v2 , but
we change Y into Y := {0, . . . , k} = k + 1. The elements of
v
Y X = (k + 1)(2)
are called labelled k–graphs , while by k–graphs on v vertices we mean the orbits of Sv on this set.
• If we want to allow loops or multiple loops, we replace X by the union v2 ∪ v, and now f (i) = j, for
i ∈ v, means that the vertex with the number i carries a j-fold loop.
4r
r
1
3r
4r
3r
@
and
@
@
@r
r
r
2
1
2
Figure 1.7: Two isomorphic labelled graphs with 4 vertices
44
Chapter 1. Classes of configurations
r
r
r
r
r
@
and
@
@
@r
r
r
Figure 1.8: The graphs obtained from the labelled ones above
• If we want to consider digraphs (i.e. the edges are directed and neither loops nor parallel edges are
allowed), we simply put
X := v 2i := {(i, j) ∈ v 2 | i 6= j},
the set of injective pairs over v.
Thus graphs, k–graphs, k–graphs with loops and also digraphs can be considered as symmetry classes of
mappings (or classes of configurations).
3
In order to apply these definitions to physical problems one has to take into account that the symmetry
of a set X, e.g. a crystal lattice, is described rather by a subgroup G < SX than by SX itself. Even the
normalizer NSX (Ḡ) in general is a nontrivial subgroup of SX and, moreover, an action G X has to preserve
an orthogonal form (the Euclidean metric in Rn ). Therefore, one has to consider the action of G < SX
on labelled graphs and to define isomorphism classes with respect to this action. It can be done by means
of corollaries 1.2.4 and 1.2.6. In the presented case (the left graph in figure 1.7) we obtain that its orbit
decomposes into three orbits under the action of D4 , which is the symmetry group of a square.
In example 1.2.13 we used Mackey’s Theorem to classify oriented pairs (x, y) ∈ X 2 and this has given a
clear classification scheme, neighbors, stabilizers etc. However, considering interactions (see 1.7.2) we have
assumed that exchange integrals are defined on non-oriented sets, in particular on
pairs. There are at least
two ways of solving this problem. At first, we can consider the action of G on X
2 (see 1.7.1) as a restriction
of the action of SX on this set of pairs. On the other hand, we can use the previously obtained results and
introduce the action of the symmetric group S2 = (1, τ ) on the orbits G(x, y) in a natural way, i.e.
S2onorb
1.7.4
τ G(x, y) = G(y, x).
It is easy to check that this definition corresponds to the action of a direct product G × S2 on mappings
f : 2 → X. After this we have to remove the the non-injective orbit G(x, x), since we want to have pairs
of different points. We will use both these approaches later on since each of them has some advantages.
It is easy to notice that in the case of symmorphic space groups the action of S2 does not lead to a new
classification scheme since it is evident that the following is true (exercise 1.7.2):
pair.cl
1.7.5 Corollary If G is a symmorphic space group with the translation group T , so that G = T o P , then:
• the stabilizer G0 (of the node labelled by 0 ∈ T ) does not contain any translations,
• all representatives of double cosets can be chosen to be translations,
• each orbit G(0, t) contains (t, 0), so G(0, t) = G{0, t}.
Since T is embedded in Rn we can characterize an orbit G{0, t} by a distance d(0, t), i.e. simply by |t| =
(t, t)1/2 (notice that all operation of a space group are isometries). It is another example of a mapping
constant on an orbit. It has to be stressed that this mapping is not injective, i.e. it is possible that two
different orbits are characterized by the same distance. Nevertheless, it enables us to introduce the notion of
the first, second, etc. neighbors of a given node. This notion can be transfered to finite lattices by means of
the introduced in section 1.6.3 epimorphism of lattices (as a distance of points in a finite lattice we choose
the shortest distance between their inverse images, for example in one dimension
d(0, N − 1) = 1 since N − 1
is the image of −1 ∈ Z). Summarizing, an action G X induces the action G X
2 , which leads to classification
1.7. Graphs and interactions
45
of pairs {x, y}. In general, the latter action is not transitive and we obtain orbits G{x, y} which, for X being
a crystal lattice, are charactezied by the distance d(x, y).
Since graphs (or configurations of exchange integrals) has been defined as mappings J: X
2 → Y then we
can consider action of G on the set of all mappings and decompose this set into orbits. Each orbit contains
configurations of exchange integrals equivalent with respect to the considered action, i.e. each orbit determines essentially different model of interaction. In the simplest case we have a configuration J with the
stabilizer GJ = G. In many case GJ < G what means that a Hamiltonian described by this configuration
breaks the symmetry of a crystal lattice. On the other hand, if we are given an action H Y it may occurs that
the stabilizer of J contains G as a proper subgroup, since J may be fixed by elements of a direct or wreath
product.
1.7.6 Heisenberg model Let us briefly consider the interaction of spins, which is a base of the so called he
Heisenberg–Dirac Hamiltonian. We have already mentioned (see 1.3.9) that the spin operator is a vector
operator with three components sx , sy , and sz . Spins interact pairwise and this interaction is given as
si · sj = sxi sxj + syi syj + szi szj ,
where each scalar operator sα
value
i , α = x, y, z, acts only at the point i of a crystal lattice Λ, i.e. on the f (i) of spin configuration. Let G be the symmetry group of Λ, so we can determine orbits G\\ Λ2 . A
Hamiltonian is invariant under the action of G if it is determined by a configuration
of exchange integrals J,
which is constant on orbits. Therefore, if T denotes any transversal of G\\ Λ2 and we assume that different
components may interact with different strengths then the most general Hamiltonian has the following form
X X
H=
J x (t)sxi sxj + J y (t)syi syj + J z (t)szi szj .
t∈T {i,j}∈G(t)
This model is simplified by some assumptions concernig both relations between J x (t), J y (t), and J z (t) for a
given t ∈ T and values of J α (t), α = x, y, z, for different orbits. For example, the condition J x (t) = J z (t) = 0
leads to Ising-like models with the Hamiltonian
X X
H=
J z (t)szi szj ,
t∈T {i,j}∈G(t)
whereas for J x (t) = J y (t) = J z (t) = J we obtain the Heisenberg Hamiltonian which, together with the
assumption that only the nearest-neighbors interact, is given be the following formula
1.7.7
H=J
X
s i · sj ,
nn
hi,ji
where hi, ji denotes a pair of nearest neighbors in
Λ
2
.
Exercise
E 1.7.1 Find representatives of three orbits into which the orbit of the first graph in figure 1.7 decomposes E
under the action of D4 .
E 1.7.2
Check statements of corollary 1.7.5.
file:f17.tex
m
46
Chapter 1. Classes of configurations
Chapter 2
Enumeration
of configuration classes
Having described the basic notions of the theory of actions of groups: orbits, stabilizers, fixed points, and
so on we discuss its enumerative aspects. We show how one can determine the number of orbits for a given
finite action on configurations. The results are based on the Cauchy-Frobenius Lemma which is applied to
the action of H o G on Y X and to its restrictions. The special cases, leading to the Involution Principle and
to some series of combinatorial numbers, are presented in detail.
Afterwards we turn to refined enumeration, where the conjugacy classes of stabilizers of orbits are prescribed.
Tables of marks and their inverses, the Burnside matrices, show up, and actions on posets and in particular
on lattices are discussed.
file:f20.tex
47
48
Chapter 2. Enumeration of configuration classes
2.1
Enumeration of configuration classes
Our paradigmatic examples are the actions of G, H, H × G, H X , and H oX G on Y X , obtained from given
actions G X and H Y . The orbits of these groups are called classes of configurations. If we want to be more
explicit, we call them G–classes, H–classes, H × G–classes, H X –classes, and H oX G–classes, respectively.
Their total number can be obtained by an application of the Cauchy-Frobenius Lemma as soon as we know
the number of fixed points for each element of the respective group. In order to derive these numbers we
characterize the fixed points of each (ψ, g) ∈ H oX G on Y X and then we use the natural embedding of G,
H, H × G, and H X in H oX G as described in 1.3.8. Thus the following lemma will turn out to be crucial:
gbar
2.1.1 Lemma Consider an f ∈ Y X , an element (ψ, g) of H oX G, and assume that
Y
(xν gxν . . . g lν −1 xν )
ḡ =
ν∈c(ḡ)
is the standard disjoint cycle decomposition of ḡ, the permutation of X which corresponds to g. Then f is a
fixed point of (ψ, g) if and only if the following two conditions hold:
• Each f (xν ) is a fixed point of the cycleproduct hν (ψ, g):
f (xν ) ∈ Yhν (ψ,g) .
• The other values of f arise from the values f (xν ) according to the following equations:
f (xν ) = ψ(xν )f (g −1 xν ) = ψ(xν )ψ(g −1 xν )f (g −2 xν ) = . . . .
This lemma yields the cycle structure of π̄ and the desired number c(π̄) of cyclic factors which we need in
order to evaluate the number of graphs on v vertices by an application of the Cauchy-Frobenius Lemma:
cycno
2.1.2 Corollary For each π̄ on
v
2
we have:
• If i is odd, then
ai (π̄) =
ai (π)
(i · ai (π) − 1) + a2i (π) +
2
X
ar (π)as (π) gcd(r, s).
r<s
lcm (r,s)=i
• If i is even, then
ai (π̄) =
ai (π)
(i · ai (π) − 2) + a2i (π) +
2
X
ar (π)as (π) gcd(r, s).
r<s
lcm (r,s)=i
• The total number of cyclic factors is
c(π̄) =
X
1X
1 X
i · ai (π)2 −
ai (π) +
ar (π)as (π) gcd(r, s).
2 i
2
r<s
i odd
Thus, by an application of the Cauchy-Frobenius Lemma, we obtain
kgraphs
2.1.3 Corollary The number of k-graphs on v vertices is equal to
X
v!−1
(k + 1)c(π̄) ,
π∈Sv
where c(π̄) is as above. More explicitly and in terms of cycle types of v (see 1.4.6) this number is equal to
X (k + 1)cā
X
1X
1 X
2
Q a
,
where
c
:=
i
·
a
−
a
+
ar as gcd(r, s).
i
ā
i
i
2 i
2
i i ai !
r<s
aàv
i odd
2.1. Enumeration of configuration classes
49
A table giving the first of these numbers looks as follows:
v\k
1
2
3
4
5
6
0
1
1
1
1
1
1
1
1
2
4
11
34
156
2
1
3
10
66
792
25506
3
1
4
20
276
10688
1601952
4
1
5
35
900
90005
43571400
Let
us apply the results of lemma 2.1.2 in order to determine the number of orbits in the action of Oh on
8
2 , i.e. we find nonequivalent pairs {j, k}, where j, k ∈ 8 are vertices of a cube. We consider this action as
a restriction of the action of S8 on this set.
2.1.4 Classification of vertex pairs At first note that some elements
of Oh have the same cycle structure oh
in S8 and, therefore, they have the same cycle structure acting on 82 . To apply the Cauchy-Frobenius Lemma
we have to find numbers of fixed points, i.e. a1 (π̄) for each π ∈ Oh (to be more precise, it is enough to find
a1 for six different cycle partitions and multiply it by the appropriate number of elements). The results are
presented below, where we give a1 (α) for different cycle partitions from the above determined table:
a1 (18 ) = 28;
a1 (32 12 ) = 1;
a1 (22 14 ) = 8;
a1 (42 ) = 0;
a1 (24 ) = 4;
a1 (62) = 1.
Therefore, the number of orbits is equal to
Oh \\ 8 = 1 (1 · 28 + 6 · 8 + 13 · 4 + 8 · 1 + 12 · 0 + 8 · 1) = 3.
2 48
Of course, these orbits are: edges, diagonals of faces, main diagonals. The number of isomorphism classes of
simple graphs on 8 cube vertices with respect to the group Oh (we need to determine ai (g) for 2 ≤ i ≤ 6) is
5 643 168, whereas using the symmetric group S8 we obtain “only” 12 346 simple graphs. Since the set of all
pairs is decomposed into three orbits then it is possible to apply results of example 1.3.11 (see also exercise
2.1.3).
3
It can be easily seen from example ?? that orbits of S2 ×S3 are formed from orbits of S2 by the action of S3 or
vice versa (we obtain a similar result considering Oh as a direct product S2 × S4 ). It is not accidental, as one
can see from the following lemma on actions of direct products, which is very easy to check (exercise 2.1.5):
2.1.5 Lemma In each case when a direct product H × G acts on a set M , we obtain both a natural action 16
of H on the set of orbits of G:
H × (G\\M ) → G\\M : (h, G(m)) 7→ G(hm),
and a natural action of G on the set of orbits of H:
G × (H\\M ) → H\\M : (g, H(m)) 7→ H(gm).
Moreover the orbit of G(m) ∈ G\\M under H is the set consisting of the orbits of G on M that form
(H × G)(m), while the orbit of H(m) ∈ H\\M under G is the set consisting of the orbits of H on M that
form (H × G)(m), and therefore the following identity holds:
|H\\(G\\M )| = |G\\(H\\M )| = |(H × G)\\M |.
In particular each action of the form H×G Y X can be considered as an action of H on G\\Y X or as an action
of G on H\\Y X . For example, considering the action S2 on orbits G\\X 2 (cf. 1.7.4) we in fact considered the
action G × S2 on X 2 and, therefore, the presented lemma yields that we can also consider the action of G on
S2 \\X 2 . The action of S2 on X 2 fixes the noninjective pairs (x, x) (they are “selfenantiomeric”), whereas the
others are grouped into “enantiomeric” pairs (x, y) and (y, x). The latter ones are in the evident bijection
with 2-subsets {x, y}.
The corresponding result on wreath products is due to W. Lehmann, and it reads as follows:
50
Chapter 2. Enumeration of configuration classes
2.1.6 Lemma The following mapping is a bijection:
Φ: H oX G\\Y X → G\\ (H\\Y )X : H oX G(f ) 7→ G(F ),
if F ∈ (H\\Y )X is defined by F (x) := H(f (x)). In particular,
|H oX G\\Y X | = |G\\ H\\Y )X |.
Exercises
E210
E 2.1.1
Check the special cases in theorem ??. In particular, show that (cf. also 2.1.6)
|H X \\Y X | = |H\\Y ||X| .
E2101
E 2.1.2 Consider an action on the set of oriented pairs, i.e. on X × X = X 2 along the line presented in
lemma ?? and corollary 2.1.2. Do the same for injective pairs, i.e. Xi2 = X 2 \ ∆(X 2 ). Determine number of
orbits for G = Oh and X = 8 as in example 2.1.4. Compare obtained results with those of exercise 1.1.5.
Mackcube
E 2.1.3 For the action of Oh on cube vertices find the number of classes of configurations for k E , k F , and
k D , where k = {0, . . . , k − 1} and the sets E, F , D conist of edges, face diagonals and main diagonals of a
cube. (Warning: There are 12 edges so you have consider the embedding Oh ,→ S12 . Note however, that
elements of Oh with the same cycle type in S8 may have — and in fact they have — different cycle types in
S12 . Of course, elements from one class in Oh have always the same type for any embedding.)
cubefaces
E 2.1.4 Consider an action of Oh on cube faces (note that they form one orbit in the action of Oh on 48 )
and compare the obtained result with the embedding S2 o S3 ,→ S6 considered in exercise 1.5.4. Determine
number of nonequivalent colorings of faces in m colors.
E1644
E 2.1.5
Check lemma 2.1.5.
E151old
E 2.1.6
Prove, by considering suitable actions, the following facts:
• ∀ n ∈ N∗ :
P
d|n
φ(d) = n.
• ∀ prime p : (p − 1)! ≡ −1 (p) (Wilson).
n
• ∀z ∈ Z, p prime, n ∈ N∗ : z (p
)
n−1
≡ z (p
)
(pn ).
• ∀z ∈ Z, n ∈ N∗ such that gcd(z, n) = 1 : z φ(n) ≡ 1 (n) (Euler).
d4spin
E 2.1.7 Determine number of nonequivalent spin configurations for the group D4 acting on square vertices.
file:f21.tex
16
2.2. The Involution Principle
2.2
51
The Involution Principle
We have evaluated the number of graphs on v vertices and the number of spin configurations for cube vertices
by examining a certain action of the form G Y X . We shall now give an example of the form H×G Y X , i.e.
a power group action. Afterwards we shall see how these two examples can be combined in order to prove
an existence theorem for a certain class of graphs (and/or) configurations. While doing so we shall meet an
interesting and useful counting principle which uses suitable actions of S2 , the smallest nontrivial group.
Investigating magnetic properties of condensed matter we also investigate the time-reversal symmetry. In
the case of spin configurations this is realized by the considertion of the following mapping
τ : [−s, +s]n → [−s, +s]n : f 7→ f˜,
∀ i ∈ n f˜(i) = −f (i).
where
It is evident that τ is a bijection and generates the group hτ i ' S2 . Moreover, for any action G [−s, +s]n
determined by the action G n we have τ (gf ) = g(τ f ), so that we can investigate the action of a direct product
hτ i × G on [−s, +s]n . We note, that the only configuration which is symmetric with respect to τ is f (i) = 0
for all i ∈ n. Of course, it exists only for integer spin number s and in this case it always forms a 1-element
orbit. Classes of spin configurations form pairs if and only if one class arises from the other by applying the
time reversion. The example of figure 2.1 shows that such classes may very well be symmetric under this
action, and hence we ask for the number of such orbits (for a given G n). It is evident that configurations in
such classes must have magnetization
X
2.2.1
M (f ) :=
f (i)
co
i∈n
equal to zero and, therefore, they are not allowed for odd n and half-integer spin number s.
t
d
t
d
d
d
t
t
t
t
d
d
t
t
d
d
Figure 2.1: Configurations forming a pair under time reversion
In order to prepare the derivation of the number of such orbits we notice that the class which we obtain by
putting a configuration class and its time-reversal image together in one class are just the orbits of the group
S2 × G on the set Y X := [−s, +s]n (recall 1.3.5). According to ?? the number of these orbits is equal to
1
2|G|
X
Y
|[−s, +s]ρi |ai (ḡ) .
(ρ,g)∈S2 ×G i∈n
But |[−s, +s]ρi |, the number of fixed points of ρi , ρ ∈ S2 , acting on the set [−s, +s], is either 2s + 1 or ,
where = 0 for half-integer spin and 1 otherwise (cf. example ?? where we have considered S2 o S3 ). We
generalize our problem and then we will be able to solve the present one as a special case.
Let Yτ denote a set of fixed points of τ acting on Y . Since τ 2 = 1 we have Yτ 2 = Y and the same holds for
all even powers of τ . Therefore, for ρ = 1, we obtain the following sum
XY
X
|Y |ai (ḡ) =
|Y |c(ḡ) ,
g
i
g
whereas for ρ = τ we have
X Y
g
i even
|Y |ai (ḡ)
Y
i odd
|Yτ |ai (ḡ) =
X
|Y |ce (ḡ) |Yτ |co (ḡ) ,
g
where co = a1 + a3 + . . . and ce = c − co . These formulas yield the following result
52
2.2.2
Chapter 2. Enumeration of configuration classes
X
1
ge
|(S2 × G)\\2n | =
|Y |c(ḡ) |Y |c(ḡ) + |Y |ce (ḡ) |Yτ |co (ḡ) .
2|G| g
In order to derive from this equation the total number of τ –symmetric configuration classes for a given G n
we use an easy argument which we have already met when we introduced the notions of enantiomeric pairs
and selfenantiomeric orbits. A group of order 2 consisting of the identity map and the involution τ acts on
the set of configuration classes. This action is chiral if |Y | > 1, and hence the desired number of τ –symmetric
classes is twice the number given in 2.2.2 minus the number of G–classes given in ??. Therefore, we can
deduce
gen.162
2.2.3 Lemma For any involution τ which commutes with the action G Y X (induced by a given action
the number of τ –symmetric orbits, i.e. such that τ G(φ) = G(τ φ) = G(φ) for all φ ∈ Y X is equal to
G X)
X |Y |ce (ḡ) |Yτ |co (ḡ)
1 X
|Y |ce (ḡ) |Yτ |co (ḡ) =
,
|G| g
|CG (g)|
g∈C
where co (ḡ) = a1 (ḡ) + a3 (ḡ) + . . ., ce (ḡ) = c(ḡ) − co (ḡ), C is a transversal of the conjugacy classes, Yτ is the
set of fixed point under the action hτ i Y , and CG (g) is the stabilizer of g ∈ G. In particular, for Yτ = ∅, we
have that this number is
1 X0 c(ḡ) X0 |Y |c(ḡ)
|Y |
=
,
|G| g
|CG (g)|
g∈C
P0
where g means the sum over all g ∈ G such that ḡ does not contain a cycle od odd lenght, i.e. co (ḡ) = 0
and ce (ḡ) = c(ḡ).
Applying this lemma to our problem we have to replace |Y | by 2s + 1 and |Yτ | by (recall that the value of
depends on the spin number), obtaining
cu.162
2.2.4 Corollary If G acts on a set of n spins s (and hence on spin configurations [−s, +s]n , too), the
number of the time-reversal symmetric classes of spin configurations is equal to


X
X
1 

(2s + 1)ce (ḡ)  ,
(2s + 1)c(ḡ) + 
|G|
g∈G
g∈G
co (ḡ)>0
co (ḡ)=0
where c(ḡ) is the total number of cycles of ḡ ∈ Sn , co (ḡ) is the number of odd cycles, ce (ḡ) = c(ḡ) − co (ḡ)
and = 0 or 1 for half-integer or integer s, respectively. In particular, since for an odd integer number each
ḡ has always at least one cycle with odd length, then for odd n and half-integer spin number s there are no
such orbits, i.e. we have
1 X
(2s + 1)c(ḡ)
2|G|
g∈G
pairs of spin configuration classes.
For example, for the octahedral group acting on vertices of a cube we obtain, using table ??,
1
(13(2s + 1)4 + 20(2s + 1)2 )
48
1
(208s4 + 416s3 + 392s2 + 184s + 33)
48
symmetric orbits for half-integer spin s and
=
1
[(13(2s + 1)4 + 20(2s + 1)2 ) + (9(2s + 1)0 + 6(2s + 1)2 )]
48
1
= (13s4 + 26s3 + 26s2 + 13s + 3)
3
otherwise. For s = 1/2 and 1 there are 6 and 27 such classes, respectively. Of course, all of them correspond
to the magnetization zero and, moreover, it occurs soon (see section 3.1) that for s = 1/2 all classes are
time-reversal symmetric, whereas for s = 1 we have 8 pairs of configuration classes under the action of hτ i.
Below we present two representatives of symmetric classes and one pair.
2.2. The Involution Principle
t
d
53
d
t
t
t
d
d
d
t
t
d
d
d
t
t
t
d
In particular each action of the form H×G Y X can be considered as an action of H on G\\Y X or as an action
of G on H\\Y X . Hence each such action of S2 × G on 2X gives rise to an action of S2 on G\\2X and leads
us to the discussion of the Involution Principle.
We call a group element τ 6= 1 an involution if and only if τ 2 = 1. The Involution Principle is a method of
counting objects by simply defining a nice involution τ on a suitably chosen set M and using the fact that
S2 ' {1, τ } has orbits of length 1 (the selfenantiomeric orbits) and of length 2 (which form the enantiomeric
pairs) only. A typical example is the time-reversion symmetry (for s > 0) or complementation τ of graphs
which is, for v ≥ 2, an involution on the set of graphs on v vertices.
We look closer at actions of involutions. The following remark is trivial but very helpful: Let τ ∈ SM be an
involution which has the following reversion property with respect to the subsets T, U ⊆ M :
m ∈ T ⇐⇒ τ m ∈ U.
2.2.5
in
Then the restriction of τ to T establishes a bijection between T and U . We shall apply this to disjoint
˙ − of M into subsets M ± . Each such disjoint decomposition gives rise to a sign
decompositions M = M + ∪M
function on M :
m ∈ M +,
sign (m) := 1
−1 m ∈ M − .
˙ − be a disjoint decomposition of a finite set M and let In
2.2.6 The Involution Principle Let M = M + ∪M
τ ∈ SM be a sign reversing involution:
∀ m 6∈ Mτ : sign (τ m) = −sign (m).
Then the the restriction of τ to M + \Mτ is a bijection onto M − \Mτ . Moreover
X
X
sign (m) =
sign (m).
m∈M
m∈Mτ
If in addition Mτ ⊆ M + , then
X
sign (m) = |Mτ | = |M + | − |M − |.
m∈M
Proof:
P
m∈M
sign (m) is equal to
X
sign (m) +
m∈Mτ
X
sign (m) .
m∈M − \Mτ
m∈M + \Mτ
|
X
sign (m) +
{z
=0,by 2.2.5
}
2
A beautiful application is provided by
2.2.7 Example Let A denote a finite set and let P1 , . . . , Pn be any given properties. We want to express ex
the number of elements of A which have none of these properties in terms of numbers of elements which
have some of these properties. In order to do this we indicate by Pi (a) the fact that a ∈ A has the property
Pi , and for an index set I ⊆ n we put
AI := {a | ∀ i ∈ I: Pi (a)}, in particular A∅ = A.
Furthermore we put
A∗ := {a | ∃
| i: Pi (a)},
54
Chapter 2. Enumeration of configuration classes
and it is our aim to express |A∗ | in terms of the |AI |. The set on which we shall define an involution is
M := {(a, I) | I ⊆ n, a ∈ AI }.
A disjoint decomposition of this set is M = M + ∪ M − , where
M + := {(a, I) | |I| even}, and M − := M \M + .
Now we introduce, for a 6∈ A∗ , the number s(a) := min{i | Pi (a)} ∈ n, and define τ on M as follows:

 (a, I ∪ {s(a)}) if a 6∈ A∗ , s(a) 6∈ I,
τ (a, I) := (a, I\{s(a)})
if a 6∈ A∗ , s(a) ∈ I,

(a, I)
if a ∈ A∗ .
Obviously 1 6= τ ∈ SM provided that A∗ 6= A. Furthermore τ 2 = 1 and τ is sign–reversing, so that from 2.2.6
we obtain
X
X
1
|A∗ | = |Mτ | = |M + | − |M − | =
1−
(a,I)∈M +
=
X
|AI | −
X
|AI | =
(a,I)∈M −
(−1)|I| |AI |.
I⊆n
|I| odd
|I| even
X
Thus we have proved
Inex
2.2.8 The Principle of Inclusion and Exclusion Let A be a finite set and P1 , . . . , Pn any properties,
while AI denotes the set of elements of A having each of the properties Pi , i ∈ I ⊆ n. Then the order of the
subset A∗ of elements having none of these properties is
X
|A∗ | =
(−1)|I| |AI |.
I⊆n
A nice application is the following derivation of a formula for the values of the Euler function φ : We would
like to express
φ(n) := |{i ∈ n | gcd(i, n) = 1}|
in terms of the different prime divisors p1 , . . . , pm of n. In order to do this we put A := n and we define
Pi , i ∈ m, to be the property “divisible by pi ”, obtaining from the Principle of Inclusion and Exclusion the
following expression for A∗ , the set of elements of n which are relatively prime to n :
A∗ = n \ {k ∈ n | pi |k, i ∈ m} ∪ {k ∈ n | (pi pj )|k, i 6= j ∈ m} \ . . . ∪ . . . ,
which yields the following formula for the cardinality of |A∗ |:
|A∗ | = n
X 1
1−
+
p
i∈m i
X
m
{i,j}∈( 2 )
1
− +...
pi p j
!
.
This gives the desired expression for the value of the Euler function at n:
Eulerx
φ(n) = n ·
2.2.9
m Y
i=1
1−
1
pi
.
3
Note that two last equalities are connected by an obvious identity for a finite set B and a mapping A: B → F,
where F is a number field,
Y
X Y
(1 + A(b)) =
A(b),
b∈B
M ∈2B b∈M
where for M = ∅ the last product is equal to 1 ex definitione. In many cases we can deal much easier with
a product of A’s than with a product of factors (1 + A). Here is a concrete example.
2.2. The Involution Principle
HTIm
55
2.2.10 High-temperature expansion Let us consider
the Ising model with (see examples 1.7.2 and
1.7.6) with exchange integrals Jij for {i, j} ∈ B = Λ2 , where Λ is the set of nodes. In the high-temperature
expansion for the partition function Z we obtain that
Y
X
(1 + Jij σi σj ).
Z∝
σ∈{−1,+1}Λ hi,ji∈B
According to the given formula we can write
X
Z∝
M ∈2B
X
Y
σ∈{−1,+1}Λ
{i,j}∈M
Jij σi σj .
Q
Note, however, that product over pairs in M leads to a monomial i∈Λ σiαi in variables σi . Therefore, the
sum over all spin configurations immediately yields that all powers αi have to be even numbers. Hence, oneand two-element1 substes M are excluded. Since B is a set of pairs then M can be interpreted as a graph. It
is easy to notice that the last condition means: M has to be a graph consisting only of cycles and, moreover,
these cycles can not have common edges.
3
Exercises
E 2.2.1
Prove ?? directly.
se
E 2.2.2 Consider an action H × S2 on Y X , for given actions
S2 on H\\Y X .
HY
and
S2 X,
as actions H on S2 \\Y X and E
E 2.2.3 Assume X to be a finite set with subsets X1 , . . . , Xn . Use the Principle of Inclusion and Exclusion E
in order to derive the number of elements of X which lie in precisely m of these subsets Xi .
E 2.2.4 Evaluate the derangement number
Dn := |{π ∈ Sn | ∀ i ∈ n: πi 6= i}|,
i.e. the number of fixed point free elementes in Sn . Derive a recursion for these numbers and show that they
tend to 1/e.
file:f22.tex
1 If
and only if |Λ| = 2 it is sometimes assumed that B is a set of oriented injective pairs, i.e. B = {(1, 2), (2, 1)}.
56
Chapter 2. Enumeration of configuration classes
2.3
Special symmetry classes
We now return to Y X and consider its subsets consisting of the injective and the surjective maps f only:
YiX := {f ∈ Y X | f injective} and YsX := {f ∈ Y X | f surjective}.
It is clear that each of these sets is both a G–set and an H–set and therefore it is also an H × G–set, but it
will not in general be an H oX G-set. The corresponding orbits of G, H and H × G on YiX are called injective
symmetry classes, while those on YsX will be called surjective symmetry classes. We should like to determine
their number. In order to do this we describe the fixed points of (h, g) ∈ H × G on these sets to prepare an
application of the Cauchy-Frobenius Lemma. A first remark shows how the fixed points of (h, g) on Y X can
be constructed with the aid of h̄ and ḡ, the permutations induced by h on Y and by g on X (use 2.1.1):
171
Q
2.3.1 Corollary If ḡ = ν (xν . . . g lν −1 xν ), then f ∈ Y X is fixed under (h, g) if and only if the following
two conditions are satisfied:
f (xν ) ∈ Yhlν ,
and the other values of f arise from the values f (xν ) according to
f (xν ) = hf (g −1 xν ) = h2 f (g −2 xν ) = . . . .
This together with 1.4.10 yields:
powerfix
2.3.2 Corollary The fixed points of (h, g) are the f ∈ Y X which can be obtained in the following way:
• To each cyclic factor of ḡ, let l denote its length, we associate a cyclic factor of h̄ of length d dividing
l.
• If x is a point in this cyclic factor of ḡ and y a point in the chosen cyclic factor of h̄, then put
f (x) := y, f (gx) := hy, f (g 2 x) := h2 y, . . . .
Such an f is injective if and only if the mapping described in the first item of 2.3.2 is injective and corresponding cyclic factors of ḡ and h̄ have the same length. The number of such mappings is
Y aj (h̄)
aj (ḡ)!,
aj (ḡ)
j
while the second item of 2.3.2 says that we have to multiply this number by Πj j aj (ḡ) in order to get the
X
number |Yi,(h,g)
| of fixed points of (h, g) on YiX . Thus we have proved
173
2.3.3 Corollary The number of fixed points of (h, g) on YiX is
Y aj (h̄)
X
|=
j aj (ḡ) aj (ḡ)!,
|Yi,(h,g)
a
(ḡ)
j
j
and hence, by restriction, the numbers of fixed points of g and of h are:
|Y | a1 (h̄)
|X|! if ḡ = 1,
X
X
|X|
and |Yi,h | =
|X|!.
|Yi,g | =
|X|
0
otherwise,
An application of the Cauchy-Frobenius Lemma yields the desired number of injective symmetry classes:
numinj
2.3.4 Theorem The number of injective H × G–classes is
|(H × G)\\YiX | =
X Y aj (h̄)
1
j aj (ḡ) aj (ḡ)!,
|H||G|
a
(ḡ)
j
j
(h,g)
2.3. Special symmetry classes
57
so that we obtain by restriction the number of injective G–classes
|X|! |Y |
|Y |
X
|G\\Yi | =
=
|SX /Ḡ|,
|X|
|Ḡ| |X|
and the number of injective H–classes
|H\\YiX | =
|Y |
|X|! X
k
|{h ∈ H | a1 (h̄) = k}|
.
|H|
|X|
k=|X|
2
In order to derive the number of surjective fixed points of (h, g) we use the preceeding corollaries together
with an application of the Principle of Inclusion and Exclusion in order to get rid of the nonsurjective fixed
points. We denote by Yν the set of points y ∈ Y contained in the ν–th cyclic factor of h̄ and put, for each
index set I ⊆ c(h̄):
X
X
Y(h,g),I
:= {f ∈ Y(h,g)
| ∀ ν ∈ I : f −1 [Yν ] = ∅}.
Then, by the Principle of Inclusion and Exclusion, we obtain for the desired number of surjective fixed points
of (h, g) the following expression:
X
X
X∗
X
|Ys,(h,g)
| = |Y(h,g)
|=
(−1)|I| |Y(h,g),I
|
I⊆c(h̄)
=
X
X
(−1)c(h̄)−|I| |Y(h,g),c(
h̄)\I |.
I⊆c(h̄)
Now we recall that
X
−1
X
[Yν ] = ∅}.
Y(h,g),c(
h̄)\I = {f ∈ Y(h,g) | ∀ ν 6∈ I : f
This set can be identified with Ye(X
, where h̃ denotes the product of the cyclic factors of h̄ the numbers of
h̃,g)
which lie in I, and where Ye is the set of points contained in these cyclic factors. Thus
Y
X
eX | =
|Y(h,g),c(
|
=
|
Y
|Yeh̃j |aj (ḡ) .
h̄)\I
(h̃,g)
j
We can make this more explicit by an application of 1.4.10 which yields:
|Yeh̃j | = a1 (h̃j ) =
2.3.5
X
d · ad (h̃).
17
d|j
Putting these things together we conclude
2.3.6 Corollary The number of surjective fixed points of (h, g) is
c(h̄)
X
|Ys,(h,g)
|=
X
(−1)c(h̄)−k
|X|
|Y | XY
ai (h̄) Y
a i=1
k=1
ai
j=1
nu

aj (ḡ)
X

d · ad 
,
d|j
where the middle sum is taken over all the sequences a = (a1 , . . . , a|Y | ) of natural numbers aj such that
P
aj = k (they correspond to all possible choices of h̃ out of h, where ai of the chosen cyclic factors of h̃
are i–cycles). Hence the numbers of surjective fixed points of g and of h amount to:
X
|Ys,g
|=
|Y |
X
(−1)|Y |−k
k=1
|Y | c(ḡ)
k ,
k
and
c(h̄)
X Y ai (h̄) |X|
(−1)
(
=
)a1 ,
ai
a
i
k=1
P
where the sum is taken over all the sequences (a1 , . . . , a|Y | ), ai ∈ N and
ai = k.
X
|Ys,h
|
X
c(h̄)−k
58
Chapter 2. Enumeration of configuration classes
An application of the Cauchy-Frobenius Lemma finally yields the desired numbers of surjective symmetry
classes:
surnum
2.3.7 Theorem The number |(H × G)\\YsX | of surjective H × G–classes is

aj (ḡ)
Y
h̄)
|Y | |X|
X c(
X
XY
X
1
a
(
h̄)
i

(−1)c(h̄)−k
d · ad 
,
|H||G|
ai
a i=1
j=1
(h,g) k=1
d|j
where the inner sum is taken over the sequences a = (a1 , . . . a|Y | ) described in the corollary above. This
implies, by restriction, the equations
|G\\YsX | =
|Y |
|Y | c(ḡ)
1 XX
(−1)|Y |−k
k ,
|G| g
k
k=1
and
|H\\YsX |
c(h̄)
X Y ai (h̄) |X|
1 XX
c(h̄)−k
(−1)
(
)a1 ,
=
ai
|H|
a
i
h k=1
where the last sum is to be taken over all the sequences a = (a1 , ..., a|Y| ) such that ai ∈ N and
P
ai = k.
These considerations lead to various combinatorial numbers so that a few remarks concerning these are in
X
is the set of surjective mappings f ∈ Y X which are constant on the c(ḡ) cyclic
order. For example, Ys,g
c(ḡ)
n
factors of ḡ. Hence this set can be identified with the set Ys . More generally we consider the set ms and
define the numbers S(n, m) by
m!S(n, m) := |mns |, where m, n ∈ N.
These S(n, m) are called the Stirling numbers of the second kind . If n is used as row index and m as column
index, then the upper left hand corner of the table of Stirling numbers of the second kind is as follows:

Stirling
2.3.8





(S(n, m)) = 




1
0
0
0
0
0
..
.

1
1
1
1
1
1
3
7
15
1
6
25
1
10





.




1
..
.
By definition m!S(n, m) is equal to the number of ordered set partitions (n(1) , . . . , n(m) ) of n into m blocks,
i.e. into m nonempty subsets n(i) . This is clear since each such ordered partition can be identified with
f : n → m where f −1 [{i}] := n(i) , i ∈ m. Thus S(n, m) is the number of set partitions {n(1) , . . . , n(m) } of the
set n into m blocks, and hence Bn , the number of all the set partitions of n satisfies the equation
Bellnu
2.3.9
Bn =
n
X
S(n, k).
k=0
These numbers Bn are called Bell numbers. Another consequence of the definition of Stirling numbers of
the second kind and 2.3.6 is
X
|Y |!S(c(ḡ), |Y |) = |Ys,g
| =
|Y |
X
k=1
which implies:
(−1)|Y |−k
|Y | c(ḡ)
k ,
k
2.3. Special symmetry classes
SFormula
59
2.3.10 Stirling’s Formula For m > 0 and any natural number n we have:
m
1 X
m−k m
S(n, m) =
(−1)
kn .
m!
k
k=1
Another series of combinatorial numbers shows up if we rewrite 2.3.7 in the following form:
|X|
2.3.11
|G\\YsX |
|Y |! X
=
S(k, |Y |)|{g ∈ G | c(ḡ) = k}|.
|G|
17
k=1
We put
r(n, k) := |{π ∈ Sn | c(π) = k}|,
and call these the signless Stirling numbers of the first kind. They satisfy the following recursion formula,
since in π ∈ Sn the point n either forms a 1–cycle or does not:
2.3.12 Lemma For n, k > 1 we have
17
r(n, k) = r(n − 1, k − 1) + (n − 1)r(n − 1, k)
while the initial values are r(0, 0) = 1 and r(n, 0) = r(0, k) = 0, for n, k > 0.
The upper left hand corner of a table containing these numbers r(n, k), for n, k ∈ N, is as follows

2.3.13





(r(n, k)) = 




1
0
1
0
1
0
2
0
6
0 24
..
.

1
3
11
50
1
6
35
1
10
1
..
17





.




.
We now return to the number |SX \\YsX |. The exercise 2.3.1 together with the identity 2.3.11 yields
|X|
2.3.14
|Y |! X
r(|X|, k)S(k, |Y |) =
|X|!
k=1
|X| − 1
.
|Y | − 1
17
Another series of combinatorial numbers arises when we count certain injective symmetry classes, since 2.3.4
implies
|Y |
17
|X|! X
k
2.3.15
|SY \\YiX | =
t(|Y |, k)
,
|Y |!
|X|
k=|X|
where t(n, k) := |{π ∈ Sn | a1 (π) = k}|. It is easy to derive these numbers from the rencontre numbers
R(n) := t(n, 0), since obviously the following is true:
2.3.16
t(n, k) =
n
R(n − k).
k
17
This can be made more explicit by an application of exercise 2.3.2 which yields
2.3.17
R(n) = n!
n
X
(−1)k
k=0
k!
.
17
60
Chapter 2. Enumeration of configuration classes
Now we use that, for |Y | ≥ |X|, |SY \\YiX | = 1, so that by the last three equations:
1=
2.3.18
|Y |
X
k=|X|
|Y |−k
X (−1)j
1
, if |Y | ≥ |X|.
(k − |X|)! j=0
j!
17
It is in fact an important and interesting task of enumeration theory to derive identities in this way since
they are understood as soon as they are seen to describe a combinatorial situation.
Another example is the identity
1717
2.3.19 Lemma For natural numbers n and k the following identities hold:
k X
k
n−1
n+k−1
1 X c(π)
k
.
=
=
n!
m m−1
n
m=1
π∈Sn
Proof: Exercise 2.3.1 implies that, for m ≤ k,
k
n−1
m m−1
is equal to the number of symmetry classes of Sn on k n , the elements of which satisfy |f [n]| = m. Thus the
left hand side is |Sn \\k n |. But the orbit of f ∈ k n under Sn is characterized by the orders of the inverse images
|f −1 [{i}]|
P , i ∈ k. Hence the number of these orbits is equal to the number of k–tupels (n1 , . . . , nk ), ni ∈ N,
and
ni = n, therefore the first identity follows from exercise 2.3.3. The last equation is already clear
from ??.
2
Exercises
n
ex171
E 2.3.1
n−1
Prove that |Sn \\ms | = ( m−1
).
ex173
E 2.3.2
Use the Principle of Inclusion and Exclusion in order to prove 2.3.17.
ex172
E 2.3.3
Show that the number of k–tupels (n1 , . . . , nk ) such that ni ∈ N and
n+k−1
.
n
ex1733
E 2.3.4
Prove that the Stirling numbers of the second kind satisfy the equation
xn =
n
X
P
ni = n is equal to
S(n, k)[x]k ,
k=0
where [x]k := x(x − 1) · . . . · (x − k + 1).
E175
k
E 2.3.5 G X is called k–fold transitive if and only if the corresponding action of G on the set of Xi of
injective mapping is transitive:
k
|G\\Xi | = 1.
Prove that, in case of a transitive action
G X,
this is equivalent to:
Gx (X\{x})
E176
E 2.3.6
is (k − 1)–fold transitive.
Prove that |G| is divisible by [|X|]k if
GX
is finite and k–fold transitive.
2.3. Special symmetry classes
E176a
61
E 2.3.7 Show that Sn is n–fold transitive on n while An is (n − 2)–fold transitive on n, but not (n − 1)–fold
transitive, for n ≥ 3.
E 2.3.8 Prove that the number of orbits for
for the action of S2 on G\\Xi2 .
X
G 2
(cf. ??) can be also determined as the number of orbits
It is probably matter of taste but for me a definition with words is more illustrative than ‘bare’ formulas.
For example, The singless Stirling numbers of the first kind provide us with the number of permutations in
Sn , which have exactly k cycles is more meaningful (for me) than only formula
r(n, k) = |{π | c(π) = k}|.
Moreover, I am so used to think with the Young diagrams that it is difficult for me to forget about them in
such cases. But the Young diagrams and associated partitions are introduced in the chapter on representations...
I haven’t changed anything in the corollary 2.3.6 and the theorem 2.3.7. I hope that you will “improve”
these formulas for the action of H taking into account that h̄ has to be identity in SY .
WSF.
file:f23.tex
62
2.4
Chapter 2. Enumeration of configuration classes
Counting by stabilizer class
e of subgroups of G. We say
Recall that the elements of an orbit have as their stabilizers a conjugacy class U
e
that U is the type of this orbit. There may be many orbits of the same type. For example, all colorings of
the regular pentagon presented in figure 1.3 belong to one of two types: monochromatic colorings with the
stabilizer D5 and the others with stabilizers conjugated to C2 = {1, σ}, where σ is defined by 1.4.15. The
e , for a given subgroup U of G, is called the U
e –stratum and indicated by
set of orbits of type U
e \\\X := {G(x) ∈ G\\X | Gx ∈ U
e }.
U
It follows immediately from lemma 1.1.16 that actions of G on all orbits in one stratum is similar, so it
is enough to carefully investigate one of these orbits to obtain results for a whole stratum. In the above
mentioned example of colorings we may limit ourselves to considerations of two, instead of eight, orbits.
The lenght of an orbit G(x) is equal to |G/Gx | and each subgroup conjugated to Gx serves as a stabilizer
for some elements in G(x). For example, if Gx is a normal subgroup of G then it is the stabilizer of each
element. The number of conjugates for a given subgroup Gx ≤ G is simply the length of the orbit G(Gx ), i.e.
fx | = |G|/|NG (Gx )| (cf. example 1.1.7). On the other hand, |NG (Gx )|/|Gx | gives the number of elements
|G
fx |
in G(x) which have Gx as their stabilizer. Threfore, we can express this number in terms of |Gx | and |G
as
1
|G(x)|
|G/Gx |
|G|
·
=
=
.
fx | |Gx |
fx |
fx |
|G
|G
|G
e \\\X then we obtain the number of x ∈ X with a
If we multiply this number by the length of a stratum U
given stabilizer U = Gx . All these elements are invariants of the action U X but not all invariants, since each
u ∈ U fixes also such y ∈ X that Gy > U . Therefore, to evaluate the cardinalty of
XU := {x ∈ X | ∀g ∈ U : gx = x},
i.e. the number of points fixed under each element of U , we have to sum up the obtained above numbers
over all subgroups V of G which include U . The resulting formula reads now
X
|XU | =
V :U ≤V ≤G
|G/V | e
|V \\\X|.
|Ve |
To write this formula in a more invariant way we introduce the zeta function
1 U ≤V
ζ: L(G) × L(G) → N: (U, V ) 7→ ζ(U, V ) :=
0 otherwise,
which describes a partial order of the lattice L(G) of all the subgroups of G. Hence, we have
|XU | =
X
V ≤G
ζ(U, V )
|G/V | e
|V \\\X|.
|Ve |
The group G is supposed to be finite so, all the more, L(G) is finite and we can number subgroups of G by
integers 1, 2, . . . , |L(G)|. Moreover, we can do this in such a way that
311
2.4.1
U i ≤ U k =⇒ i ≤ k
(i.e. we can embed the partial order into a total one) what allows to introduce the zeta–matrix
ζ(G) := (ζ(U i , U k ))
which is upper triangular, with ones along its main diagonal. Using this matrix we can rewrite the obtained
formula as a linear problem




..
..
.
.
i



e i \\\X| 
.
 |XU i |  = ζ(G) ·  |G/Ui | |U


 |Ue |

..
..
.
.
2.4. Counting by stabilizer class
63
To solve this problem, i.e. to express the length of a stratum in terms of the number of invariants, it is
enough to determine the inverse of ζ(G) (it is obvious that ζ(G) is invertible). This new matrix, called the
Moebius–matrix,
µ(G) := (ζ(G))−1 = (µ(U i , U k ))
is also an upper traingular matix over integers with ones along its main diagonal. The corresponding mapping
µ: L(G) × L(G) → N: (U, V ) 7→ µ(U, V ),
is called the Moebius fuction on L(G). It helps to write the formula for the desired length of the stratum of
U:
X
e
31
e \\\X| = |U |
2.4.2
µ(U, V )|XV |.
|U
|G/U |
V ≤G
This close connection between the zeta and the Moebius function provides a very useful inversion theorem
which we introduce next. It is given in the simplified version for finite posets, but it can be easily generalized.
The proof is left as exercise 2.4.1.
2.4.3 The Moebius Inversion Let (P, ≤) denote a finite poset and let F and G denote mappings from P M
into the field F. Then we have the following equivalence of systems of equations:
X
X
∀ p : G(p) =
F (q) ⇐⇒ ∀ p : F (p) =
G(q)µ(q, p),
∀ p : G(p) =
q≤p
q≤p
X
X
F (q) ⇐⇒ ∀ p : F (p) =
q≥p
µ(p, q)G(q).
q≥p
The partial order ≤ in (P, ≤) allows us to introduce intervals
[p, q] := {r ∈ P | p ≤ r ≤ q}
and half-open intervals
[p, q) := {r ∈ P | p ≤ r < q},
(p, q] := {r ∈ P | p < r ≤ q}.
It is easy to prove the following relation (exercise 2.4.2)
X
µ(p, r) =
r∈[p,q]
X
µ(r, p) = δpq =
r∈[p,q]
1
0
if p = q
otherwise,
which yields the recursion formulas
2.4.4
µ(p, q) = −
X
r∈[p,q)
µ(p, r) = −
X
µ(r, q).
22
r∈(p,q]
It easy to note that in the equation 2.4.2 only the Moebius function depends on a subgroup V , while the
other numbers depend on a conjugacy class (of U or V ). The length of an orbit, |G/U |, is evidently constant
e and the same regards the number of fixed points since the following mapping is a
on a conjugacy class U
bijection:
2.4.5
XU → XgU g−1 : x 7→ gx.
ad
e
Let us consider the set L(G)
consisting of the conjugacy classes of subgroups of G:
e
e1 , . . . , U
ed }, with representatives Ui ∈ U
ei .
L(G)
:= {U
(U i should not be mixed up with Ui !) Putting
bi
64
Chapter 2. Enumeration of configuration classes
2.4.6
bik :=
ei | X
|U
µ(Ui , V ), and B(G) := (bik ),
|G/Ui |
ek
V ∈U
we can now reformulate 2.4.2 in terms of this matrix B(G), called the Burnside matrix of G (this term
has been introduced by A. Kerber). Actually, Burnside considered the inverse of B(G) and called it the
table of marks; we shall return to this matrix later. It is in fact the following lemma (and not the lemma
of Cauchy–Frobenius that is quite often named after Burnside, see the remarks on history in the chapter
containing the comments and references) which is
Blemma
2.4.7 Burnside’s Lemma If
G on X satisfies the equation
GX
ei \\\X| of the strata of
is a finite action, then the vector of the lengths |U



..
..
.
.




e



 |U
 i \\\X|  = B(G) ·  |XUi |  .
..
..
.
.

We can apply this now in order to enumerate the G–classes on Y X by type. Since f ∈ Y X is fixed under
each g ∈ Ui if and only if f is constant on each orbit of Ui on X, we obtain for finite G Y X :
312
ei is the i-th entry of the one column
2.4.8 Corollary The number of classes of G-configurations of type U
matrix


..
.


|Ui \\X| 
B(G) · 
|Y
|

.
..
.
d4lattice
2.4.9 Example According to 1.4.16 the dihedral group D4 is generated by π = (1 2 3 4) and σ = (1 3).
This group has 10 subgroups which, using these generators, can be written as follows:
U 1 = h1i ' S1 , U 2 = π 2 ' C2 ,
U 3 = hπσi ' S20 , U 4 = π 3 σ ' S20 , U 5 = hσi ' S2 , U 6 = π 2 σ ' S2 ,
U 7 = π 2 , πσ ' C2 × S20 = D20 , U 8 = π 2 , σ ' C2 × S2 = D2 ,
U 9 = hπi ' C4 , U 10 = hπ, σi ' D4 ,
where we have also put the isomorphic images of these subgroups (note that S2 for U 5 , U 6 , and U 8 acts on
the set 4 in a different way than for U 3 , U 4 , and U 7 and therefore one of these groups is distinguished by
“prime”). It is not difficult to draw the lattice of subgroups which looks as follows (the conjugate classes
are marked by ovals):2
U 10 s
AHH
A HHH
HHs U 8
AAs U 7
U9 s
A AA
A
A
A
U2 s
U 3 s As U 4 U 5 s As U 6
!
!
@
@
!!
!
@ !!
@
s
!!
1 U
2 Please
choose one of these pictures. I like this asymmetric one, since it puts S2 -isomorphic group aside from the C2 group.
However, the ‘symmetric’ version is more clear. WSF
2.4. Counting by stabilizer class
65
U 10 s
SS
S
U 7 s U 9 s
Ss U 8
S
S
AA
S A
U 3 s
s U 4 Ss 2 U 5 s As U 6
c
#
#
c SS U #
c
S #
c
Ss
#
c
U1
Since it is very easy to construct the zeta-matrix, so we present the Moebius–matrix only:


1





µ(D4 ) = 





−1
1
−1
0
1
−1
0
0
1
−1
0
0
0
1
−1
0
0
0
0
1
2
−1
−1
−1
0
0
1
2
−1
0
0
−1
−1
0
1
0
−1
0
0
0
0
0
0
1
0
2
0
0
0
0
−1
−1
−1
1





.





The enumeration of classes of spin-configurations with respect to the dihedral group D4 by type needs the
Burnside matrix of this group, which can be easily obtained by applying 2.4.6 and the previous one. The
reslut looks as follows:


1/8 −1/8 −1/4 −1/4 1/4 1/4
0
0

1/4
0
0 −1/4 −1/4 −1/4 1/2 



1/2
0 −1/2
0
0
0



1/2
0 −1/2
0
0
.
B(D4 ) = 

1/2
0
0 −1/2 



1/2
0 −1/2 



1/2 −1/2 
1
This matrix has to be applied to the vector of numbers of configurations fixed by each of the subgroups Ui ,
which is
 

(2s + 1)4
16s4 + 32s3 + 24s2 + 8s + 1
2
2
4s + 4s + 1
 (2s + 1)  

 


4s2 + 4s + 1
 (2s + 1)2  




3
3
2
8s + 12s + 6s + 1
 (2s + 1)  


.
1=
2s + 1
 (2s + 1)  




2
2
4s + 4s + 1
 (2s + 1)  




1
(2s + 1)
2s + 1
1
(2s + 1)
2s + 1

The product of the Burnside matrix and this vector turns out to be
 4

2s + 2s3 − s2 /2 − s/2
0




2s2 + s




4s3 + 4s2 + s



.
0




2s2 + s




0
2s + 1
The sum of all entries should be equal to the result of exercise 2.1.7. In table 2.1 we present numbers of
orbits counted by type for few smallest spin numbers s = 0, 1/2, 1, 3/2, 2, 5/2.
66
Chapter 2. Enumeration of configuration classes
One can immediately notice two important features. At first, the spin number s = 1/2 is too small to form
asymmetric (regular) orbit, but it is clear from the table and the general formula that the number of such
orbits, i.e. the length of the stratum e
1\\\[−s, +s]4 , increases very fast and for large spin numbers almost all
e2 ,
configurations are asymmetric. The second, even more important fact, is that subgroups in the classes U
e
e
U5 , and U7 may never be stabilizers in the considered case. In the other words, if a configuration is fixed
by, for example, the cyclic group C4 then, in a sense automatically, it is fixed by D4 .
3
The above presented example stimulates a general question: Which subgroups of G occur as stabilizers of
mappings f ∈ Y X ? In order to answer this question let us consider the stabilizer of f ∈ Y X in SX . Evidently,
it is equal to the following direct sum of the symmetric groups on the inverse images of the y ∈ Y :
stabsx
(SX )f =
2.4.10
M
SXy , where Xy := f −1 [{y}].
y
This implies
stabcond
2.4.11 Corollary The stabilizers of the f ∈ Y X are the subgroups U ≤ G, the corresponding permutation
group Ū of which satisfies
Ū = Sα ∩ Ḡ,
where Ḡ corresponds to G, and Sα is the direct sum of the symmetric groups on the orbits of U :
M
Sα :=
Sω .
ω∈U \\X
For example, the action of C4 on 4 leads to one orbit, so Sα = S4 and, therefore, Sα ∩ D4 = D4 6= C4 . In
this way we have partially proved the following
dunstab
2.4.12 Corollary The cyclic group Cn does not occur as a stabilizer of spin configurations on a regular
polygon if n ≥ 3. Moreover, if n = 2m is an even integer then also the cyclic group Cm may not be a
stabilizer of spin configurations. The only stabilizer of the order n is the dihedral group Dm genrated by π 2
and σ, where π and σ a given by 1.4.15.
Proof: The first statement is clear. The cyclic group Cm decomposes n into two orbits consisting of odd
and even numbers, respectively. The dihedral group D2m contains the permutation σ = (1 2m−1)(2 2m−
2) . . . (m−1 m+1)(m)(2m) (cf. 1.4.15) which belongs to S{1,3,...,2m−1} ⊕ S{2,4,...,2m}
. Therefore,
σ ∈ Sα ∩ D2m
and then the last cross-section is different from Cm . This also proves that Dm = π 2 , σ may be a stabilzer.
If we consider the other group isomorphic with Dm generated by π 2 and πσ then it is easy to notice that
πσ
2 = (1 2m)(2 2m−1) . . . (m−1 m+2)(m m+1) mixes odd and even integers, so in this case Sα = Sn and
π , πσ may not be
a stabilizer
of any spin configurations. Since Dn has three normal subgroups of the
order n (hπi, π 2 , σ , and π 2 , πσ ) then the proof is completed.
2
Table 2.1: Spin configurations on a square, counted by type
s \ type
0
1/2
1
3/2
2
5/2
e1
U
0
0
3
15
45
105
e2
U
0
0
0
0
0
0
e3
U
0
1
3
6
10
15
e4
U
0
2
9
24
50
90
e5
U
0
0
0
0
0
0
e6
U
0
1
3
6
10
15
e7
U
0
0
0
0
0
0
e8
U
1
2
3
4
5
6
P
1
6
21
55
120
231
2.4. Counting by stabilizer class
67
A similar result holds for the cyclic group Cv and the alternating subgroups Av−1 and Av acting on multigraphs (see exercises 2.4.4 and 2.4.5).
The set
ek (G, X)
of all the classes of stabilizers (or epikernels as we shall call them) for a given action G X is a subposet of the
e
poset L(G)
of all the conjugacy classes of subgroups. For example, structural phase transitions characterized
e belongs to ek (G, X) for an
by a group–subgroup relation G ≥ U are addmissible if and only if the class U
appropriate (as a rule geometric) action of G — a symmetry group of the prototypic (disordered) phase.
Another immediate consequence of 2.4.2 is an expression for the number of orbits of prescribed length. Using
the abbreviation, for U, V ≤ G:
X
31
2.4.13
µ(U, W ),
µ(U, Ve ) :=
e
W ∈V
we derive from 2.4.2
2.4.14 Corollary If
|G\\k X|, is equal to
GX
is a finite action, then the number of orbits of length k of G on X, denoted 31
1
k
X
ei |
|U
X
i:|G/Ui |=k
ej )|XU |,
µ(Ui , U
j
j
and in particular the number of asymmetric orbits, i.e. of orbits of type e
1 or of maximal length |G|, is equal
to
1 X
ej )|XU |.
|G\\|G| X| =
µ(1, U
j
|G| j
2.4.15 Example Let us determine the number of pairs, i.e. orbits of the length two. It follows from pa
corollary 2.4.14 that such orbits have as stabilizers subgroups with the index 2, so these subgroups are
ei | = 1. Moreover, they are maximal in G, i.e. the only nonzero values
normal in G. Therefore, in this case |U
of the Moebius function in 2.4.13 are obtained for V = U and V = G. As the result we obtain that there are
X
1
(|XU | − |XG |)
2
U :|G/U |=2
2-element orbits, where XG is the set of invariants of the action G X. In the considered case of the action of
0
D2m on [−s, +s]2m it is easy to notice that C2m and Dm
fix only (2s + 1) constant mappings, as also D2m
does, while Dm — with two orbits on 2m — fixes (2s + 1)2 ones. Therefore, in this case we have
2
(2s + 1) − (2s + 1) /2 =
2s + 1
2
pairs. They, of course, correspond to different values of spin projections on each of two orbits of Dm acting
on 2m, i.e. for odd and even numbers, respectively. One of these pairs contains a mapping
n
s for i odd
fNéel (i) =
−s for i even
which corresponds to the so-called Néel configuration.
3
The formula obtained in this example resemblances a bit relations between enantiometric pairs and selfenantiometric orbits. It is not surprising since each (normal) subgroup of the index two determines the chirality
of an action. It is evident that considering a restriction of G X to one of these subgroups U , say, we obtain
fixed points instead of two-element orbits. Pairs which are related to other subgroups remain, since their
elements are transformed into each other by, among others, elements of this chosen U . If we restrict ourselves
68
Chapter 2. Enumeration of configuration classes
e being an union of one- and two-element orbits then, assuming G+ = U , we obtain the following
to the set X
formulas (cf. 1.4.21)
X
e =1
|G\\X|
(|XV | − |XG |) + |XG |,
2
V :|G/V |=2
e =
|U \\X|
And, therefore, for the action
1
2
X
(|XV | − |XG |) + |XU |.
V :|G/V |=2
V 6=U
GX
e we have
e − |U \\X|
e = |XG | +
2|G\\X|
1
2
X
(|XV | − |XG |)
V :|G/V |=2
V 6=U
selfenatiometric orbits and
e − |G\\X|
e = 1 (|XU | − |XG |) .
|U \\X|
2
enantiometric pairs (what can be easily pressumed) with respect to the chirality determined by U C G (and
e We do not want to go into detailed discussion but a similar analysis can be done for other
on the set X).
orbits after that we can classify enatiomeric pairs and selfenatiomeric orbits by type. We compare only the
obtained classification of orbits by type for D4 with this for D2 (note that the conjugate class of S2 in D4
decomposes into two classes in D2 and hence we have doubled the entry in this row; rows with two zeros are
omitted):
type \ group
D4
D2
4
3
2
4
S1
2s + 2s − s /2 − s/2
4s + 4s3 + s2
S20
2s2 + s
0
3
2
3
4s + 4s + s
2 · (4s + 4s2 + s)
S2
2
2s + s
4s2 + 4s + 1
D2
D4
2s + 1
0
This table yields the following classification scheme: orbits of types S1 , S2 , and D2 consist of enantiomeric
pairs which splite over D2 ; the other orbits of types S20 and D4 are selfenatiometric, so they do not split (but
their types are changed, of course).
This is very first – so very rough — attempt to this problem. It should be corrected or removed. WSF
The corollary 2.4.11 yields that for an action of Sn on mn , induced by the natural action of Sn on n, all
stabilizers have to be given as direct sums (cf. also 1.1.5)
M
Sα =
Sn(i)
i∈m
S
determined by a decompostion n = ˙ i n(i) of n into m subsets. (Note that some these subsets can be empty,
so it is not a set-partition.) The conjugacy class of such subgroups can be labelled by a partition α ` n with
no more than m parts (so we can identify ek (Sn , mn ) with the set of such partitions). The corresponding
decomposition is given as follows
i−1
i
X
X
(i)
n = {1 +
αj , . . . ,
αj }
j=1
j=1
and it determines a represenative of one orbit in this stratum. Representatives of the other orbits can be
obtained by the action of σ ∈ Sm (note that not all σ lead to a new orbit). This raises a question about
relations between counting orbits by type and the action of a direct product Sm × Sn on mn . The answer
can be formulated as
Weydualp
2.4.16 Weyl duality for permutations
To be more concrete we assume that n is a set of crystal
nodes (or particles), while m is a set of one-node states (e.g. the set of spin projections [−s, +s] which can
bijectively mapped onto m by [−s, +s] 3 j 7→ j + s + 1 = i ∈ m). The action of Sn and its subgroup G is
often referred to as the place permutations (or particle permutations). The second action, of H ≤ Sm , is
known as state permutations since it results from permutations on the set of single-spin states. We know
2.4. Counting by stabilizer class
69
that these actions commutes, so we can consider the action of H × G on mn . This condition can be looked
at as the Weyl duality between the group G acting on the set n and the group H acting on m. Recalling the
discucssion on fiber bundles in section 1.6 we can also say that it is the duality between permutation groups
G and H acting on the base n and the typical fiber m, respectively.
The first remark is due to lemma 2.1.5: the action of H × G on mn can be considered as an action of H on
e \\\mn , U ≤ G, is an invariant subset under the action of
G\\mn and vice versa. Moreover, each stratum of U
n
H. The same is true for the action of G on H\\m (exercise 2.4.7). Therefore, we can consider subactions of
e \\\mn ,
G on all strata Ve \\\mn , with Ve ∈ ek (H, mn ) and, on the other hand, subactions of H on all strata U
e ∈ ek (G, mn ). If an orbit ω ∈ U
e \\\mn then the action G ω is similar to the action G G/U . It means
where U
e \\\G is similar to the cartesian product
that the action of the direct product H × G on a stratum U
G G/U
e \\\G).
× H (U
Since the set mn is a disjoint union of the strata then we obtain the following similarity
M
n
e
G G/U × H (U \\\G).
G×H m '
e∈ek(G,mn )
U
In the same way we can obtain the dual formulation reads
M
n
e
G×H m '
G (V \\\H) × H H/V.
e ∈ek(H,mn )
V
If the action of G (or H) on a given stratum is not transitive we can consider its decomposition into transitive
actions and in this way obtain description of the action in question in terms of transitive actions.
3
Exercises
E 2.4.1
Prove the Moebius Inversion 2.4.3 for finite posets.
M
E 2.4.2
Check the recusions 2.4.4.
M
E 2.4.3
Prove the following identity for the elements of the Burnside matrix:
E
X
e
ei , Uk ) |Uk | , where µ(U
ei , Uk ) :=
bik = µ(U
µ(V, Uk ).
|G/Ui |
ei
V ∈U
E 2.4.4 Prove that the cyclic group Cv does not occur as an automorphism group of a multigraph on v E
vertices, if v ≥ 2.
E 2.4.5 Prove that the group Av and its subgroup Av−1 do not occur as automorphism groups of a E
multigraph on v vertices, if v ≥ 3.
E 2.4.6
Classify selfenantiomeric orbits and enantiomeric pairs by type with respect to the chirality
determined by C4 and D20 for the action of D4 on spin configurations.
E 2.4.7 Show that any stratum Ve \\\mn , V ≤ H ≤ Sm , is invariant under the action of G ≤ Sn and vice st
versa, i.e. that strata of G acting on mn are invariant under H.
file:f24.tex
70
2.5
Chapter 2. Enumeration of configuration classes
Tables of marks and Burnside matrices
We have seen the significance of the Burnside matrix for the enumeration of orbits by type. The inverse of
this matrix was introduced by Burnside (1911) and called the table of marks . Sometimes it is also called
the supercharacter table for a reason which will become clear later. We denote this matrix by
M (G) := (mik ) := B(G)−1 ,
and state:
321
2.5.1 Lemma
mik =
e , V ) :=
where ζ(U
P
e
W ∈U
|G/Uk |
ek ) = |G/Uk | ζ(U
ei , Uk ) ∈ N,
ζ(Ui , U
e
ei |
|Uk |
|U
ζ(W, V ) and ζ(U, Ve ) is defined in a corresponding way.
Proof: We start by proving the last of the two identities. It follows directly from the equation
zyx
ei |ζ(Ui , U
ek ) = |U
ek |ζ(U
ei , Uk ),
|U
2.5.2
an identity which is immediate from the definition of the zeta function. The first of the stated identities is
now obtained as follows:
X
j
bij
X
e
|G/Uk |
ek ) = |Ui |
ek )
ej ) |G/Uk | ζ(Uj , U
ζ(Uj , U
µ(Ui , U
e
ek |
|G/Ui | j
|Uk |
|U
=
ei ||Ui | X X X
|U
µ(Ui , V )ζ(V, W ) = δik .
ek ||Uk |
|U
ek j V ∈Uej
W ∈U
|
{z
}
=δUi ,W
The final claim that mik ∈ N follows from the first identity since the order of a conjugacy class of a subgroup
is equal to the index of the normalizer, thus
323x
2.5.3
|G/Uk |
|NG (Uk )|
=
∈ N.
e
|Uk |
|Uk |
2
This lemma shows that the entries of M (G) describe the poset
e
(L(G),
)
ei of subgroups, 1 ≤ i ≤ d, together with the partial order
which consists of the conjugacy classes U
3222
2.5.4
ei U
ek :⇐⇒ ∃ U ∈ U
ei , V ∈ U
ek : U ≤ V.
U
Lemma 2.5.1 implies that
3233
2.5.5
ei U
ek .
mik 6= 0 ⇐⇒ U
Burnside called M (G) the table of marks for the following reason:
marks
2.5.6 Lemma The entry mik is the number of left cosets of Uk in G which remain fixed under left multiplication by the elements of Ui .
2.5. Tables of marks and Burnside matrices
71
Proof:
|{gUk | ∀x ∈ Ui : xgUk = gUk }| =
=
|G|
ei |
|Uk ||U
1
|{g ∈ G | g −1 Ui g ≤ Uk }|
|Uk |
ei | V ≤ Uk }| =
|{V ∈ U
|G/Uk | e
ζ(Ui , Uk ) = mik .
ei |
|U
2
Hence mik is, so to speak, the mark which Ui leaves when it is acting on the left cosets of Uk . We now
derive further properties of these elements. As G is assumed to be finite, we can choose a numbering of the
ei such that the following holds:
conjugacy classes U
|Ui | < |Uk | =⇒ i < k.
2.5.7
32
This guarantees in particular that the partial order is respected:
ei U
ek =⇒ i ≤ k.
U
2.5.8
32
Under the assumptions of 2.5.7, M (G) is upper triangular, U1 = {1}, while, since d is the number of
conjugacy classes of subgroups, Ud = G, and so the table of marks takes the following form:

|G| . . .
|G/Uk |
... 1
.
..

.

∗ .. 

.. 
M (G) = 
|NG (Ui )/Ui |
.
.



.
..

. .. 
0
1

2.5.9
ta
Other consequences of 2.5.1 are divisibility properties:
ei U
ek , then mik = mkk ζ(Ui , U
ek ), and so mkk divides all the mik in the same column. 32
2.5.10 Lemma If U
Moreover, certain nonzero elements in a column form a monotonous sequence, for mik 6= 0 6= mjk means
ei U
ek U
ej . If in addition U
ei U
ej holds, which implies that i ≤ j, then we have
that U
ei U
ej U
ek .
U
As we may assume Ui ≤ Uj without restriction, we obtain:
gUj g −1 ≤ Uk =⇒ gUi g −1 ≤ Uk ,
so that an application of
2.5.11
mik =
1
|{g ∈ G | gUi g −1 ≤ Uk }|
|Uk |
32
finally yields
ei U
ej U
ek , then the corresponding elements in the k-th column of the table of 32
2.5.12 Corollary If U
marks of G satisfy
mik ≥ mjk ≥ mkk .
72
Chapter 2. Enumeration of configuration classes
The evaluation of the table of marks is usually quite difficult since one has to know L(G), its Hasse3 diagram,
and the orders of the Ui and their normalizers. Fortunately there exists the Aachen subgroup lattice program
by V. Felsch, incorporated also in the program system CAYLEY, which for example was used to evaluate
the tables in the Appendix.4
Burnside’s original motivation for introducing the table of marks was the problem of decomposing a given
action into its orbits or, in other words, to decompose a permutation representation into its transitive
constituents. The question was whether it suffices to consider only the character of the action G X, i.e. the
function χ: g 7→ |Xg |. In order to explain theoretically what is meant by the decomposition of the action
G X into its transitive constituents we first recall that there exists a natural equivalence relation on the set
of actions of G on finite sets. Two actions, G X and G Y , say, were called similar if and only if there exists a
bijection ϕ: X → Y which is G–invariant,, i.e. for which the following holds:
∀ x ∈ X, g ∈ G : ϕ(gx) = gϕ(x).
We know from 1.1.16 that there are exactly as many similarity classes of transitive actions as there are
conjugacy classes of subgroups. An immediate implication is
3211
2.5.13 Lemma The set { G (G/Ui ) | 1 ≤ i ≤ d} is a transversal of the similarity classes of transitive actions
of G.
Proof: The actions G (G/Ui ) and G (G/Uk ), i 6= k, are not similar, since the respective stabilizers are Ui and
Uk , which are different.
2
3212
2.5.14 Corollary The characters
χi : G → C: g 7→ |(G/Ui )g |,
ei , 1 ≤ i ≤ d, are the transitive characters of G. These characters have the
of the actions G (G/Ui ), Ui ∈ U
following values (recall that C G (g) denotes the conjugacy class and CG (g) the centralizer of g):
χi (g) =
|G| |C G (g) ∩ Ui |
.
|Ui | |C G (g)|
Proof: Consider a transversal T of the left cosets of Ui . By definition of χi , we have
χi (g) = |{t ∈ T | gtUi = tUi }| = |{t ∈ T | t−1 gt ∈ Ui }|
= |{g 0 ∈ G | g 0 gg 0
−1
∈ Ui }|
|C G (g) ∩ Ui |
1
=
|CG (g)|,
|Ui |
|Ui |
and the result follows since |G/CG (g)| = |C G (g)|.
2
Burnside saw that a knowledge of the character χ of G X together with a table of the χi does not suffice
to decompose χ into its transitive constituents. Such a decomposition is equivalent to the evaluation of the
coefficients ni ∈ N in the equation
d
X
χ=
ni χi ,
i=1
where ni is the number of orbits of G on X similar to
G (G/Ui ),
as G–sets.
Later on we shall provide an example which shows that this linear combination is not uniquely determined.
But it turns out that replacing the χi by the rows of the table of marks we get a unique linear combination.
We first show that the χi form part of the table of marks.
3214
2.5.15 Lemma If Ui = hgi is a cyclic subgroup of G and χk the character of
3I
G (G/Uk ),
don’t what it is. WSF
I said, I am going to prepare an appendix containg tables for S2 o S3 , i.e. Oh , group. WSF
4 As
then mik = χk (g).
2.5. Tables of marks and Burnside matrices
73
Proof: By an application of 2.5.11 we obtain
mik =
1
−1
|{g 0 ∈ G | g 0 gg 0 ∈ Uk }| = χk (g).
|Uk |
2
This is the reason why M (G) is sometimes called the table of supercharacters of G, and it also shows that it
is helpful to indicate the columns of M (G) which belong to cyclic subgroups so that we can easily identify
the transitive characters from M (G). For example the table of marks of S4 is

2.5.16








M (S4 ) = 







24
12
2
12
.
4
8 6 6 6 4 3
. 2 . . 2 1
. 2 2 6 . 3
2 . . . 1 .
2 . . . 1
2 . . 1
6 . 3
1 .
1
2
.
2
2
.
.
2
.
.
2

1
1

1

1

1

1

1

1

1

1
1
←
←
←
←
32
←
a table which corresponds to the following numbering of subgroups Ui :
U1 = h1i, U2 = h(24)i, U3 = h(13)(24)i, U4 = h(132)i,
U5 = h(13), (24)i, U6 = h(1234)i, U7 = h(12)(34), (14)(23)i,
U8 = h(132), (13)i, U9 = h(1234), (24)i, U10 = h(132), (142)i,
U11 = h(1324), (1342)i.
The rows which correspond to cyclic groups are marked by an arrow ←, and hence the table of transitive
permutation characters of S4 is as follows:
χ1
χ2
χ3
χ4
χ5
χ6
χ7
χ8
χ9
χ10
χ11
(14 )
24
12
12
8
6
6
6
4
3
2
1
(12 2)
.
2
.
.
2
.
.
2
1
.
1
(22 )
.
.
4
.
2
2
6
.
3
2
1
(13)
.
.
.
2
.
.
.
1
.
2
1
(4)
.
.
.
.
.
2
.
.
1
.
1
As M (G) is upper triangular if the numbering of the conjugacy classes of subgroups satisfies 2.4.1, also
B(G), its inverse, is upper triangular. Furthermore several of its entries can be made explicit in terms of the
Moebius function of the lattice of subgroups L(G) (recall 1.1.9):
2.5.17 Corollary If the numbering of the conjugacy classes of subgroups of G satisfies 2.4.1, then we have 32
for the Burnside matrix of G:
 1

µ(1,Uk )
µ(1,G)
. . . |N
...
|G|
|G|
G (Uk )|


..
..


.


∗
.


|Ui |
µ(Ui ,G)
B(G) = 
.
|NG (Ui )|
|NG (Ui )/Ui | 



.
..
..


.
0
1
74
Chapter 2. Enumeration of configuration classes
From the definitions of mik and bik we obtain interesting relations between various elements or products of
elements of these two matrices, e.g. that the following products are rational integral:
3219
ek )µ(Uk , U
ej ) ∈ Z.
mik bkj = ζ(Ui , U
2.5.18
Exercises
ex321
E 2.5.1
Show that the number of generators of a cyclic group of order n is equal to φ(n).
file:f25.tex
2.6. Actions on posets, semigroups, lattices
2.6
75
Actions on posets, semigroups, lattices
We have discussed the Burnside matrix and the table of marks. They belong (in a sense which will become
clear later) to the subgroup lattice L(G) on which G acts by conjugation:
G × L(G) → L(G): (g, U ) 7→ gU g −1 .
2.6.1
ac
This operation respects the partial order ≤ of L(G):
U ≤ U 0 =⇒ gU g −1 ≤ gU 0 g −1 ,
2.6.2
41
and it respects the operations ∧ and ∨ on L(G):
g(U ∧ U 0 )g −1 = g(U ∩ U 0 )g −1 = gU g −1 ∧ gU 0 g −1 ,
g(U ∨ U 0 )g −1 = ghU ∪ U 0 ig −1 = gU g −1 ∨ gU 0 g −1 .
We therefore consider actions of finite groups which respect a partial order or a multiplication or both the
operations ∧ and ∨ of a finite lattice. In other words, we are going to consider actions of finite groups as
groups of automorphisms of posets, semigroups and lattices, in the following sense. Let (X, ≤) denote a
poset. The group G acts on X as a group of automorphisms if and only if the following holds:
∀ g ∈ G, x, x0 ∈ X (x ≤ x0 ⇐⇒ gx ≤ gx0 ).
2.6.3
pa
We shall abbreviate this by writing
G (X, ≤),
and we shall call this a poset action.
2.6.4 Lemma If
G (X, ≤)
denotes a finite poset action, then the following holds:
• Any two elements in the same orbit are incomparable, i.e. the orbits are antichains.
• The orbits ωi of G on X can be numbered in such a way that
ωi 3 x ≤ x0 ∈ ωk =⇒ i ≤ k.
• For any orbit ω and a fixed x ∈ X, the numbers
|{x0 ∈ ω | x ≤ x0 }| and |{x0 ∈ ω | x ≥ x0 }|
depend only on the orbit containing x and not on the chosen representative.
• For any x, x0 ∈ X, we have
|{x00 ∈ X | x ≤ x00 ∈ G(x0 )}|
|G(x0 )|
=
.
000
0
000
|{x ∈ X | x ≥ x ∈ G(x)}|
|G(x)|
Proof:
i) If x ∈ X were comparable with gx 6= x, say (without restriction) x < gx, then we have
x < gx < g 2 x < . . . < g −1 x < x,
which is a contradiction.
ii)Suppose x1 , x2 ∈ ωi , x01 , x02 ∈ ωk , i 6= k. Assume that x1 < x01 and that x2 and x02 are comparable. Then
x2 > x02 would yield, for suitable g, g 0 ∈ G: gx1 = x2 > x02 = g 0 x01 , and hence also x1 > g −1 g 0 x01 , which
41
76
Chapter 2. Enumeration of configuration classes
s
s
s
x000
x0
s ...
x00
s
s ... s
x
s
s
G(x0 )
s
G(x)
Figure 2.2: Two orbits of the poset
implies the contradiction x01 > g −1 g 0 x01 . Thus the partial order can be embedded into a total order that
respects the partial order:
x1 , x2 , . . . , x|ω1 | , x|ω1 |+1 , . . . , x|ω1 |+|ω2 | , . . . , where xi < xk ⇒ i < k.
{z
} |
{z
}
|
∈ ω1
∈ ω2
iii) is clear from 2.6.3
iv) follows from a trivial “double count”. We consider the bipartite graph consisting of the two orbits G(x)
and G(x0 ), where comparable elements are joint by an edge (see Figure 2.2).
2
A lattice (L, ∧, ∨) defines a poset (L, ≤) and besides this it yields two semigroups (L, ∧) and (L, ∨), as both
these compositions are associative by the definition of a lattice.
416
2.6.5 Lemma Let (L, ∧, ∨) denote a lattice on which a finite group G acts. The following three conditions
are equivalent:
• ∀ x, x0 , g : x < x0 =⇒ gx < gx0 ,
• ∀ x, x0 , g : g(x ∧ x0 ) = gx ∧ gx0 ,
• ∀ x, x0 , g : g(x ∨ x0 ) = gx ∨ gx0 .
Proof:
i)⇒ ii): As x ∧ x0 is less than or equal to both x and x0 , we obtain from i) that g(x ∧ x0 ) is less than or equal
to both gx and gx0 . This yields
g(x ∧ x0 ) ≤ gx ∧ gx0 .
(?)
If now g(x ∧ x0 ) were strictly less than gx ∧ gx0 , we also had, by i):
g 2 (x ∧ x0 ) < g(gx ∧ gx0 ) ≤ g 2 x ∧ g 2 x0 ,
where the last inequality comes from (?). Hence, for each n ∈ N, we obtain
g n (x ∧ x0 ) < g n x ∧ g n x0 ,
which yields a contradiction by putting n := |hgi|. Thus g(x ∧ x0 ) = gx ∧ gx0 must hold.
i)⇒ iii) follows quite analogously.
ii)⇒ i): The assumption x < x0 yields x ∧ x0 = x, so that g(x ∧ x0 ) = gx, and hence, by ii), gx ≤ gx0 . This
implies gx < gx0 , as x 6= x0 .
iii)⇒ i) follows similarly.
2
Lemma 2.6.5 means that we may either check if the action respects the partial order or one of the two
compositions ∧ or ∨. In each of these cases we shall use either the notation G (L, ≤) or we shall indicate this
situation by
G (L, ∧, ∨)
and call such actions lattice actions. An immediate consequence of 2.6.4 is
2.6. Actions on posets, semigroups, lattices
aik
2.6.6 Corollary If
numbers
G (L, ∧, ∨)
77
denotes a finite lattice action, then, for any orbit ω and a fixed x ∈ L, the
|{x0 ∈ ω | x ≤ x0 }| and |{x0 ∈ ω | x ≥ x0 }|
depend only on the orbit containing x and not on the chosen representative x. Thus, having numbered the
orbits in the way described in 2.6.4, we can introduce the numbers
0
0
∨
0
0
a∧
ik := |{x ∈ ωk | x ≤ x }| and aik := |{x ∈ ωk | x ≥ x }|,
∧
∧
∨
where x is an element of ωi . The a∧
ik form an upper triangular matrix A := (aik ), while the aik form a
∨
∨
lower triangular matrix A := (aik ). The main diagonals consist of ones, and hence both these matrices are
invertible over Z. Furthermore the elements of these matrices are related by the equations
∨
|ωi | · a∧
ik = |ωk | · aki .
Now we consider the more general case, where a group G acts as a group of automorphisms on a semigroup
(X, ·), i.e. we assume that the action also satisfies
∀ x, x0 , g : g(x · x0 ) = gx · gx0 .
Such actions are called semigroup actions and they are indicated by
G (X, ·).
We denote the orbits of G on X by
ω1 , . . . , ω d .
A trivial but important remark is (exercise 2.6.1):
2.6.7 Lemma For each i, j, k ∈ d and any z, z 0 ∈ ωk we have
l4
|{(x, x0 ) ∈ ωi × ωj | x · x0 = z}| = |{(x, x0 ) ∈ ωi × ωj | x · x0 = z 0 }|.
In other words: The number of solutions (x, x0 ) ∈ ωi × ωj of x · x0 = z, z ∈ ωk , does not depend on the chosen
z but only on its orbit. We can therefore denote this number by
2.6.8
a·ijk := |{(x, x0 ) ∈ ωi × ωj | x · x0 = z}|,
for a fixed z ∈ ωk . (The reader should note the upper index “·” which indicates the multiplication in
question, and which therefore is not a fly blow.)
We next introduce a ring which has these numbers as its structure constants. To do this we start from the
semigroup ring of X over Z, which is the set
ZX = {f | f : X → Z},
together with addition and multiplication defined by:
(f + f 0 )(x) := f (x) + f 0 (x), (f ? f 0 )(x) :=
X
f (x0 )f 0 (x00 ).
x0 ·x00 =x
We denote the resulting ring by ZX,· . Its elements will be written as “formal sums”
X
f=
fx x, where fx := f (x).
x∈X
If, in addition, we are given an action G X, then we call the f that are fixed under each g ∈ G, for short:
the f ∈ ZX
G , the G–invariants or the G–invariant mappings (recall the notation XG introduced in the first
section). They form an important subring (exercise 2.6.2), the main properties of which are gathered in the
following lemma:
ai
78
Chapter 2. Enumeration of configuration classes
2.6.9 Lemma
su
• The G–invariants f ∈ ZX form the subring:
−1
ZX,·
}.
G := {f : X → Z | ∀g ∈ G: f = f ◦ ḡ
• This subring has as a Z–basis the orbit sums
ωi :=
X
x ∈ ZX,·
G .
x∈ωi
·
• The structure constants of ZX,·
G are the aijk defined in 2.6.8, i.e. we have for the product of basis
elements
X
ωi · ωj =
a·ijk ωk .
k
In this way each action of a finite group G on a finite lattice (L, ∧, ∨) as a group of lattice automorphisms
yields the two rings ZL,∧ and ZL,∨ together with their subrings
ZL,∧
and ZL,∨
G
G ,
the multiplicative structure of which is described by the constants
0
0
a∧
ijk := |{(x, x ) ∈ ωi × ωj | x ∧ x = z}|,
0
0
a∨
ijk := |{(x, x ) ∈ ωi × ωj | x ∨ x = z}|,
for a fixed z ∈ ωk .
A paradigmatic example is formed by the subgroup lattice L := L(G) and the action of G on it by conjugation.
We are now in a position to state and prove the main theorem of this section (W. Plesken):
Plesken
2.6.10 Theorem Let G (L, ∧, ∨) denote a finite lattice action of G. Assume that ω1 , . . . , ωd are the orbits,
ω1 , . . . , ωd their sums, numbered according to 2.6.4. Then
• The mapping

a∧
1k


ωk 7→  ... 

a∧
dk
defines a ring isomorphism between ZL,∧
and Zd , while
G
• the mapping

a∨
1k


ωk 7→  ... 

a∨
dk
defines a ring isomorphism between ZL,∨
and Zd ,
G
where Zd is equipped with pointwise addition and multiplication.
∧
Proof: In order to check the homomorphy first, we consider
product a∧
i · aj of the i–th and j–th column
P ∧ the
∧
∧
∧
∧
of A . We want to verify that it satisfies ai · aj = k aijk ak . From the definition of a∧
li we obtain (for a
fixed x ∈ ωl ):
∧
a∧
li · alj = |{y ∈ ωi | y ≥ x}| · |{z ∈ ωj | z ≥ x}|
X
∧
= |{(y, z) ∈ ωi × ωj | x ≤ (y ∧ z)}| =
a∧
ijk alk ,
k
2.6. Actions on posets, semigroups, lattices
79
∨
∨
which proves homomorphy. In order to check the isomorphy we use that both A∧ := (a∧
ik ) and A := (aik )
are triangular by 2.6.6 and contain only 1’s along their main diagonal (and hence are invertible over Z). This
shows that the above mappings are even Z–isomorphisms, which completes the proof of the first statement,
the second one follows analogously.
2
These results show the fundamental importance of the columns of A∧ . But the rows also have interesting
properties (see exercise 2.6.3).
Exercises
E 2.6.1
Prove 2.6.7.
E
E 2.6.2
Check the first statement of 2.6.9.
E
∧
∧
E 2.6.3
Prove that the matrices A∧
as eigenvectors. What are the E
i := (aijk ) have the rows of A
eigenvalues?
file:f26.tex
80
Chapter 2. Enumeration of configuration classes
Chapter 3
Weights
Now we refine our methods in order to enumerate orbits with prescribed properties. We introduce a weight
function on G X, i.e. a mapping from X into a commutative ring which is constant on the orbits of G on X, and
the Cauchy-Frobenius Lemma will be refined in order to count orbits with prescribed weight. For example
we shall be able to enumerate graphs by their number of edges or spin configurations by magnetization.
This leads us to certain generating functions, the so–called cycle indicator polynomials. The enumeration
of rooted trees amounts to the consideration of sums of cycle indicator polynomials and leads to recursive
methods. These recursions can be used even for constructive methods, and they stimulate the formalization
of the calculation of generating functions. Next we show the join classification: by stabilizer class (i.e. by
type) and by weight together with the corresponding generating functions.
file:f30.tex
81
82
3.1
Chapter 3. Weights
Enumeration by weight
In the preceding sections we saw that various structures like graphs or spin configurations can be defined as
orbits of groups on sets in a natural way so that we already have a method at hand to evaluate the total
number of such structures by an application of the Cauchy-Frobenius Lemma. The question arises how this
lemma can be refined in such a way that we can also derive the number of orbits with certain prescribed
properties like, for example, the number of graphs on v vertices which have e edges or the number of spin
configurations with prescribed magnetization. We solved this problem in the case when property means type.
In this section we will provide a more general answer introducing a weight which mostly will be a mapping
from the set on which the group is acting into a polynomial ring over Q. The final result will be a generating
function for the enumeration problem in question, i.e. we shall obtain a polynomial which has the desired
numbers of orbits as coefficients of its different monomial summands. The basic tool is
CFW
3.1.1 The Cauchy-Frobenius Lemma, weighted form Let G X denote a finite action and w: X → R a
mapping from X into a commutative ring R containing Q as a subring. If w is constant on the orbits of G
on X, then we say that w is a weight function and we have, for any transversal T of the orbits:
X
1 X X
1 X X
w(x) =
w(x).
w(t) =
|G|
|Ḡ|
g∈G x∈Xg
t∈T
ḡ∈Ḡ x∈Xḡ
This result implies 1.1.12 (put w: x 7→ 1), which we shall sometimes call the constant form of the CauchyFrobenius Lemma.
How can we use this lemma in order to evaluate the number of orbits with prescribed properties? Note
that the result of the calculation is the sum of all the weights of the elements in a transversal T of all the
orbits. Hence we can extract from this sum the number of orbits of prescribed weight if and only if the
different weights are linearly independent. Hence, to begin with, we determine the set of all the properties
of the orbits which we want to take into account. Now, to each of these properties we assign an element
of the ring R, a weight. The ring R usually is a polynomial ring, and therefore we mostly associate with
each property a monomial. And then, if we associated different monomials with different properties, then
the various monomials, i.e. the various weights, are linearly independent, and therefore we can read from
the result of our calculation the coefficient of the corresponding monomial. This coefficient is precisely the
number of orbits with that weight.
In order to apply 3.1.1 to the enumeration of symmetry classes of mappings f in Y X we introduce, for a given
mapping W : Y → R, R being a commutative ring with Q as a subring, the multiplicative weight w: Y X → R,
defined by
Y
w(f ) :=
3.1.2
W (f (x)),
x∈X
and notice that for any finite actions
GX
and
HY
the following is true:
3.1.3 Corollary If W is constant on the orbits of H on Y , then w is constant on the orbits of H oX G, H ×
G, H and G on Y X . Moreover, for any W , the corresponding multiplicative weight function w is constant
on the orbits of G on Y X .
Thus 3.1.1 can be applied as soon as we have evaluated the sum of the weights of those f which are fixed
under (ψ, g) ∈ H oX G. But this sum of weights follows directly from the characterization of the fixed points
of (ψ, g) given in 2.1.1:
3.1.4 Corollary Using the same notation as in 2.1.1, for (ψ, g) in H oX G, a function W : Y → R constant
on the orbits of H, and the corresponding multiplicative weight w: Y X → R, we obtain the equation
X
Y X
w(f ) =
W (y)lν .
X
f ∈Y(ψ,g)
ν y∈Yhν (ψ,g)
Now an application of the weighted form of the Cauchy-Frobenius Lemma to 3.1.3 yields the desired generating function for the enumeration of symmetry classes by weight:
3.1. Enumeration by weight
83
3.1.5 Theorem Let G X and H Y be finite actions, W : Y → R a mapping into a commutative ring containing
Q as a subring, and denote by w the corresponding multiplicative weight function on Y X .
• If W is constant on the orbits of H on Y , then w is constant on the orbits of H oX G on Y X , and, for
each transversal T of these orbits, we have
X
t∈T
w(t) =
c(g)
1
Y
X
|H||X| |G|
X
W (y)lν .
(ψ,g)∈HoXG ν=1 y∈Yhν (ψ,g)
Moreover w is also constant on the orbits of H × G and H, so that, by restriction, we obtain for the
sum of the weights of the elements in a transversal the expressions
1
|H||G|
and
X
|X|
Y
ai (ḡ)

X

(h,g)∈H×G i=1
W (y)i 
,
y∈Yhi

|X|
1 X X
W (y) .
|H|
h∈H
y∈Yh
• For any W : Y → R the corresponding multiplicative weight function w: Y X → R is constant on the
orbits of G on Y X , and the sum of its values on a transversal of the orbits is equal to

ai (ḡ)
|X|
X
Y
X
1

W (y)i 
.
|G|
i=1
g∈G
y∈Y
In several cases we shall use an additive weight instead, i.e. a mapping
X
v: Y X → R: f 7→
V (f (x)),
x∈X
where V : Y → R is a mapping into commutative ring containing the rational numbers again. It can be easily
transformed into a multiplicative weight introducing
W : Y → Q[u]: y 7→ uV (y) ,
where Q[u] is a polynomial ring over Q in the indeterminate u. This mapping yields the multiplicative weight
Y
W (f (x)) = uv(f ) .
w(f ) =
x∈X
This solution is, in general, good but in many cases we cannot define ur , where r ∈ R and, moreover, such
expressions are not polynomials since r is not a natural, not even an integral number. However, introducing
more indeterminates, say u1 , . . . , um , we can deal with this problem. For example, let R be a polynomial
ring over Q in the indeterminate t. Then we can introduce indeterminates u0 , u1 , . . . and determine the
mapping W : Y → Q[{u}] as
Y q (y)
W (y) =
ui i ,
i
where qi (y) are coefficients of a polynomial V (y) = i qi (y)ti . It is still no good if qi (y) are not integers.
We can again enlarge our set of indeterminates introducing uij for each possible denominator j ∈ N of
q (y)
jq (y)
coefficients qi (y), i.e. we replace ui i
by uij i . We can do even better if we can determine the least
common multiplicity q of all denominators, since we need not to introduce new indeterminates but we
q (y)
qq (y)
simply replace ui i
by ui i , where all the products qqi (y) are integer numbers.
This approach can be
√
easily applied when t is the immaginary unit i or such irrational numbers like 2 or π. The last step which
we can do (and we have to do if W (y) is to be a monomial with non-negative powers of the indeterminates
P
84
Chapter 3. Weights
u0 , u1 , . . .) is to “shift” all exponnents by the minimal negative value of qqi (y) for each i (this minimum
is taken over all y ∈ Y ). It is not necessary if all exponents are positive integer numbers but also can be
applied to obtain monomials W (y) with the smallest possible degree. If pi denotes these minima then the
final form of W is as follows
Y qq (y)−p
i
W (y) =
ui i
i
and it yields the multiplicative weight
w(f ) =
Y
W (f (x)) =
Y
q
ui
P
x
qi (f (x))−|X|pi
=
Y
,
i
i
x∈X
qvi (f )−|X|pi
ui
where vi (f ) is the i-th coefficient in the additive weight
X
X X
v(f ) =
V (f (x)) =
qi (f (x)) ti .
i
x∈X
x∈X
This “shift” has, however, one disadvantage since w(f ) depends on |X| and we have to be very careful
comparing results for different sets X (e.g. considering the size effects for spin systems with a given spin
number s and lattices consisting of |Λ| = 2, 4, 6, . . . nodes).
magwe
3.1.6 Example In 2.2.1 we defined the magnetization of a configuration as
X
f (i), where f (i) ∈ [−s, +s] ⊂ Q.
M (f ) =
i∈n
In this case we can use the fact that for a given spin number s differences between any values of f (i) are
integer. Moreover, only for an odd n and a half-integer s we obtain M (f ) being half-integer for all f .
Therefore, we can simply define
W1 (j) = uj , j ∈ [−s, +s],
or, if we want to avoid half-integer exponents, as
W2 (j) = u2j ,
which yields the multiplicative weight
w2 (f ) =
Y
W2 (f (i)) = u2M (f ) .
i∈n
However, this leads to polynomials with neagtive exponents of u. We can avoid this introducing
W : [−s, +s] → Q[u]: W (j) = uj+s .
This yields that
w(f ) =
Y
W (f (i)) = uM (f )+ns .
i∈n
As we mentioned above, such weight depends on n and, for example, M (f ) = 0 corresponds to different
monomials uns for different sets n.
3
The most general weight function is obtained when we take for W a mapping which sends each y ∈ Y to a
separate indeterminate of a polynomial ring. For the sake of notational simplicity we can do this by taking
the elements y ∈ Y themselves as indeterminates and putting W : Y → Q[Y ]: y 7→ y, where Q[Y ] denotes
the polynomial ring over Q in the set Y of commuting indeterminates. This yields the multiplicative weight
w(f ) = Πx f (x), a monomial in Q[Y ]. If we define the content of f ∈ Y X to be the mapping
3.1.7
c(f, −): Y → N: y 7→ |f −1 [{y}]|,
i.e. c(f, y) is the multiplicity f takes the value y with, then we get
3.1. Enumeration by weight
c
Z
c Zc
B
Bc
c
85
s
Z
Zc
c
B
Bc
c
s
Z
c Zs
B
Bc
c
s
Z
Zc
c
B
Bc
s
Figure 3.1: One half of the different necklaces
3.1.8 Corollary The number of G−classes on Y X , the elements of which have the same content as f ∈ Y X ,
is equal to the coefficient of the monomial Πy y c(f,y) in the polynomial
1
|G|
|X|
X Y
ai (g)

X

g∈G i=1
yi 
.
y∈Y
A nice example is the so–called necklace problem:
3.1.9 Example We ask for the different classes of necklaces with n beads in up to m colours and with given
content, where two necklaces belong to the same class if and only if the first one can be obtained from the
second one by a suitable rotation. Hence a necklace is a coloring
f ∈ mn .
Hence we are faced with an action of the form G Y X , namely the natural action of the cyclic group G := Cn
on the set Y X := mn . (In case we want to allow reflections, we have to consider G := Dn , the dihedral
group.) A particular case was already discussed in 1.4.14, where we obtained the total number of orbits in
the special case when n is prime. Now we are in a position to count these orbits by content. In order to
do this for general m and n we take from 1.3.2 the cycle structure of the elements of Cn and obtain, by an
application of 3.1.8, the desired solution of the necklace problem:
3.1.10 Corollary The number of different necklaces containing bi beads of the i–th colour, i ∈ m, is the
bm
in the polynomial
coefficient of y1b1 . . . ym
1X
d n/d
φ(d)(y1d + . . . + ym
) ,
n
d|n
where φ denotes the Euler function (see 1.3.2).
For a numerical example we take m := 2 and n := 5 and obtain the generating function
1
((y1 + y2 )5 + 4(y15 + y25 )) = y15 + y14 y2 + 2y13 y22 + 2y12 y23 + y1 y24 + y25 .
5
Recall that the monomial summand 2y13 y22 means that there are exactly two different necklaces consisting of
5 beads three of which are of the first colour and two of which are of the second colour. Figure 3.1 shows
four of the eight different necklaces, the remaining ones are obtained by simply exchanging the two colours.
3
3.1.11 Example Let us consider orbits of the action Oh [−s, +s]8 (cf. ??). At first note that according oh
to lemma 2.3.19 the polynomial, which we are going to obtain, will have 8+2s
terms and the sum of
8
their coefficients should be equal to the number given by ??. For s = 1 these numbers are 45 and 267,
respectively, so below we present only the case s = 1/2. To simplify notation we assume that W (−1/2) = m
and W (+1/2) = p, so the derived polynomial belongs to Q[{m, p}] and looks as follows (cf. table ??)
1
(m + p)8 + 6(m + p)4 (m2 + p2 )2 + 8(m + p)2 (m3 + p3 )2
48
86
Chapter 3. Weights
+13(m2 + p2 )4 + 8(m2 + p2 )(m6 + p6 ) + 12(m4 + p4 )2 .
= m8 + m7 p + 3m6 p2 + 3m5 p3 + 6m4 p4 + 3m3 p5 + 3m2 p6 + mp7 + p8 .
Obtained results confirm our presumptions: there are 9 terms and the sum of coefficients equals 22. Moreover,
the content c(f, p) = c(f, m) = 4 corresponds to configuartions with the magnetization 0, so all these orbits
are symmetric under the time-reversion (cf. corollary 2.2.4). Below there are configurations which represent
orbits with two m and six p. Number of these orbits is given by the coefficient 3 of the monomial m2 p6 . Of
course, due to time-reversal symmetry, they have their counter-parts decribed by the monomial 3m6 p2 .
t
t
d
t
t
t
t
t
d
t
t
t
d
t
t
d
d
t
t
t
t
t
t
d
3
3.1.12 Example In the last example we show an application of the Cauchy-Frobenius Lemma to an action
of the finite group D4 = C2 o S2 , where C2 = {1, −1} on the infinite set Z2 . A generating function which we
shall obtain is written as a an inifite series. The group C2 o S2 acts on Z2 as the point symmetry group of a
square crystal lattice. As a weight we choose monomials in Q[u] determined by the absolute value of k ∈ Z,
i.e. W (k) = u|k| . This yields the multiplicative weight w(k1 , k2 ) = u|k1 |+|k2 | to be the length of (k1 , k2 ) in
the city metric. (One can also consider also a square of the Euclidean metric d(k1 , k2 )2 = k12 + k22 but it
leads the number theoretic problem of solutions of the Diophantine equations k12 + k22 = n, n ∈ Z.) All
cycleproducts can be only ±1 and it is evident that
Y1 = Z and Y−1 = {0}.
The first leads to a series
X
u|k|l ,
k∈Z
where l is the length of a cycle factor of σ ∈ S2 , while the second gives the constant term u0 . The latter one
is the only term for elements ((−1, −1), 1), ((1, −1), (12)), and ((−1, 1), (12)). The elements ((1, −1), 1) and
((−1, 1), 1) yields the following series
∞
X
X
un .
u|k| = u0 + 2
k=1
k∈Z
For ((1, 1), (12)) and ((−1, −1), (12)) in the same way we obtain
u0 +
∞
X
u2n .
k=1
The idenity ((1, 1), 1) is a bit cubersome, since we have
X X
u|k1 |+|k2 | =
k1 ∈Z k2 ∈Z
∞
X
un
n=0
= u0 + 2
n
X
X
1
k1 =−n k2 :|k2 |=n−|k1 |
∞
X
un (1 +
n=1
n−1
X
k=−n+1
1) = u0 + 4
∞
X
nun .
n=1
Therefore, the Cauchy-Froebenius Lemma 3.1.1 yields
X
t∈T
w(t) = u0 +
∞
∞
X
1X n
u (n + 1) + u2n =
(n + 1)(1 + u)u2n .
2 n=1
n=0
From this result we can find that for given n there are (n+1) orbits containing elements with w(k1 , k2 ) = u2n
and there are exactly the same number of orbits with w(k1 , k2 ) = u2n+1 . For example, when n = 2 we have
3 orbits represented by (4,0), (3,1), and (2,2) with the weight u4 and 3 orbits with representatives (5,0),
(4,1), (3,2) and weight u5 .
3.1. Enumeration by weight
87
Besides the Cauchy-Frobenius Lemma also the Involution Principle admits a generalization to a weighted
form:
3.1.13 The Involution Principle, weighted form Assume that τ is a sign-reversing involution acting
˙ − and that w: X → R is a weight function which is constant on the orbits of τ .
on the finite set X = X + ∪X
Then
X
X
sign (x)w(x) =
sign (x)w(x).
x∈X
x∈Xτ
The proof is trivial but the applications are not.
Exercises
E 3.1.1 Prove the following combinatorial principle: If X and Y are finite sets and R is a commutative
ring, and ϕ: Y × X → R, then
Y X
X Y
ϕ(f (x), x) =
ϕ(y, x).
f ∈Y X x∈X
x∈X y∈Y
E 3.1.2 Derive 3.1.8 directly, using the fact that f ∈ Y X is fixed under g ∈ G if and only if f is constant
on the cyclic factors of ḡ.
file:f31.tex
88
Chapter 3. Weights
3.2
Cycle indicator polynomials
We have seen in 3.1.8 that the generating function for the enumeration of the G–classes on Y X by weight is
equal to

ai (ḡ)
|X|
1 X Y  X i
y
.
|G|
i=1
g∈G
y∈Y
This polynomial can be obtained from the polynomial
C(G, X) :=
|X|
1 X Y ai (g)
∈ Q[z1 , . . . , z|X| ]
zi
|G|
i=1
g∈G
P
by simply replacing the indeterminate zi by the polynomial y y i . It is therefore the polynomial C(G, X)
which really matters. We call this polynomial the cycle indicator polynomial or the cycle index of G X, since
it displays the cycle structure of the elements ḡ ∈ Ḡ. (But it should be mentioned that C(G, X) = C(H, X)
does not mean that Ḡ and H̄ are isomorphic. There are counterexamples known: for example, the regular
representation of the nonabelian group of order p3 , p an odd prime number, has the same cycle indicator
polynomial as the regular representation of the abelian group Cp × Cp × Cp (exercise 3.2.3.) In the case when
we wish to display its indeterminates we shall write C(G, X; z1 , . . . , z|X| ), and if we replace zi by ri ∈ Q, we
simply write C(G, X; r1 , . . . , r|X| ). Hence we obtain, for example (cf. ??):
polconst
3.2.1 Corollary The number of G–classes on Y X is equal to
C(G, X; |Y |, . . . , |Y |).
In order to generalize the substitution process mentioned above which yields the generating function from
C(G, X), we put, for any polynomial p in Q[u1 , . . . , um ],
C(G, X | p(u1 , . . . , um )) := C(G, X)|zi :=p(ui1 ,...,uim )
=
|X|
1 XY
p(ui1 , . . . , uim )ai (ḡ) ∈ Q[u1 , . . . , um ],
|G|
i=1
g∈G
and call this process Pólya–substitution. (Note that in fact there are two substitutions: for a given i =
1, . . . , |X| we at first substitute uj , j ∈ m, by uij and next each zi by the previously obtained polynomial.)
Using this notation we can rephrase corollary 3.1.8 as follows:
Polya
3.2.2 Pólya’s Theorem The generating function for the enumeration
of G–classes on Y X by content can
P
be obtained from the cycle indicator of G X by Pólya–substituting y∈Y y into the cycle indicator polynomial
C(G, X). Hence this generating function is equal to
X
C(G, X |
y) = C(G, X)|zi :=Σyi .
y∈Y
In order to count G–classes by content it therefore remains to evaluate the cycle indicator of
examples should be welcome:
ex223
3.2.3 Examples of cycle indicator polynomials
• The cycle indicator of the identity subgroup of Sn and its natural action on n is
C({1}, n) = z1n .
• The cycle indicator of the natural action of the cyclic group Cn on n is (recall 3.1.10)
C(Cn , n) =
1X
n/d
φ(d)zd .
n
d|n
G X.
A few
3.2. Cycle indicator polynomials
89
• The cycle indicator of the natural action of the dihedral group Dn of order 2n on the set n (which can
be considered as being the set of vertices of the regular n–gon, of which Dn is the symmetry group)
satisfies, if n ≥ 3,
(
(n−1)/2
1
1
z z2
if n is odd,
C(Dn , n) = C(Cn , n) + 12 1 n/2
(n−2)/2
2
2
+ z1 z2
) if n is even.
4 (z2
This follows from the fact that the rotations form a cyclic subgroup of order n, while the remaining
reflections either leave exactly one vertex fixed (in the case when n is odd) or two vertices or none (in
the case when n is even), and group the remaining vertices into pairs of vertices that are mapped onto
each other.
• Furthermore we have the following cycle indicators of the natural actions of Sn and An (cf. 1.4.6 and
exercise 1.5.2):
X Y 1 zk ak
,
C(Sn , n) =
ak ! k
aàn k
C(An , n) =
X
(1 + (−1)a2 +a4 +... )
aàn
Y 1 zk ak
.
ak ! k
k
3
Let 8 be the set of cube vertices (cf. example ??) and [−s, +s] = {−s, −s + 1, . . . , s} the set of possible spin
projections for a given spin number s. It is our aim to evaluate number of classes of spin configurations on
the cube with prescribed properties.
3.2.4 Spin configurations on the cube The cycle indicator polynomial follows from the table ?? and sp
we obtain
1 8
C(Oh , 8) =
(z + 6 z14 z22 + 8 z12 z32 + 13 z24 + 8 z2 z6 + 12 z42 ).
48 1
From this polynomial we can derive a lot of results. To begin with, we can deduce the total number of classes
of configurations in the case when there are 2s + 1 = |[−s, +s]| possible spin projections. By replacing each
zi by 2s + 1 we obtain that the number of nonequivalent configurations is equal to (cf. ??)
n(s) =
1
(16s8 + 64s7 + 136s6 + 184s5 + 181s4 + 130s3 + 66s2 + 21s + 3).
3
These numbers for s = 0, 1/2, 1, . . . , 7/2 are gathered in a table below:
s
n(s)
0
1
1/2
22
1
267
3/2
1 996
2
10 375
5/2
41 406
3
135 877
7/2
384 112
This result can be refined by enumerating the classes of configurations by weight and using Pólya’s theorem.
At first let us consider the magnetization, i.e. the weight defined in example 3.1.6. The generating function
is given as
X
C(Oh , 8 |
uj+s ).
j∈[−s,+s]
In the resulting polynomial with the indeterminate u the coefficient of uM +8s provides a number of desidered
configurations. The first two nontrivial polynomials (for s = 1/2, 1) are given below:
s = 1/2
s=1
u8 + u7 + 3u6 + 3u5 + 6u4 + 3u3 + 3u2 + u + 1,
u16 + u15 + 4u14 + 6u13 + 15u12 + 19u11 + 32u10 + 34u9
+43u8 + 34u7 + 32u6 + 19u5 + 15u4 + 6u3 + 4u2 + u + 1,
Note that for s = 1/2 orbits with a given magnetization have also exactly determined content since
c(f, +1/2) = M + 4. Moreover, we clearly see that the number of orbits depends only on the absolute
value of M . In the case of antiferromagnetic interactions we are looking for the ground state in the subspace
90
Chapter 3. Weights
with M = 0, and the above formulas yield that there are 6 and 43 such orbits. For larger spins (up to s = 4)
these numbers are as follows:
236,
979,
3 262,
9 199,
22 800,
51 059.
Later on we shall see how one can obtain a transversal of these orbits having this one for orbits with M = −1
(in general, we shall show recursive methods of generation of a transversal for a given M having a transversal
for M −1).
The number of orbits symmetric under the time-reversion can be easily obtained by the substitution zi = 2s+1
for even i and zi = otherwise, where = 0 or 1 for half-integer or integer s, respectively (see lemma 2.2.4).
Of course, it is true for any spin system, so if G is the symmetry group of a crystal lattice Λ then the number
of symmetric configurations is equal to
C(G, Λ; , 2s + 1, , 2s + 1, . . .).
In the considered case, i.e. G = Oh and Λ = 8, we have
1
(13(2s + 1)4 + 20(2s + 1)2 )
48
symmetric spin configurations for half-integere s, while for integer s we must add to this number
1
(9 + 6(2s + 1)2 )
48
what we obtain after substituting z1 = z3 = 1.
The most general weight, i.e. the subsitution
zj =
s
X
yij
i=−s
provides the most detailed information but the derived polynomials are the longest ones. Since for s = 1/2
the classification is identical with that by the magnetization, then we present the result for s = 1 only. The
summands are grouped according to the magnetization, so it is easy to compare the derived polynomial with
the previous one (y1̄ = y−1 ):
y1̄8 + y1̄7 y0 + 3y1̄6 y02 + y1̄7 y1 + 3y1̄5 y03 + 3y1̄6 y0 y1
+ 6y1̄4 y04 + 6y1̄5 y02 y1 + 3y1̄6 y12 + 3y1̄3 y05 + 10y1̄4 y03 y1 + 6y1̄5 y0 y12
+ 3y1̄2 y06 + 10y1̄3 y04 y1 + 16y1̄4 y02 y12 + 3y1̄5 y13
+ y1̄ y07 + 6y1̄2 y05 y1 + 17y1̄3 y03 y12 + 10y1̄4 y0 y13
+ y08 + 3y1̄ y06 y1 + 16y1̄2 y04 y12 + 17y1̄3 y02 y13 + 6y1̄4 y14
+ y07 y1 + 6y1̄ y05 y12 + 17y1̄2 y03 y13 + 10y1̄3 y0 y14
+ 3y06 y12 + 10y1̄ y04 y13 + 16y1̄2 y02 y14 + 3y1̄3 y15
+ 3y05 y13 + 10y1̄ y03 y14 + 6y1̄y 20 y15 + 6y04 y14 + 6y1̄ y02 y15 + 3y1̄2 y16
+ 3y03 y15 + 3y1̄ y0 y16 + 3y02 y16 + y1̄ y17 + y0 y17 + y18
Recall that the coefficient 17 of y1̄3 y02 y13 means that there exist exactly 17 configurations with 3 projections
+1, 2 projections equal to 0 and 3 projections –1. Note that this polynomial reflects the time-reversal
symmetry since it is invariant under the transposition y1̄ ↔ y1 .
3
The next example presents an application of the Cauchy-Frobenius Lemma in the weighted form to counting
(simple) graphs.
3.2. Cycle indicator polynomials
91
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
@
@r
r @
r
r
@
@r
r @
r
r
@
@r
r @
r
r
@
@r
r @
r
r
@
@r
r @
Figure 3.2: The graphs on 4 points
3.2.5 Example For the graphs on 4 vertices we use the cycle index (exercise 3.2.1)
221w
3.2.6
C(S4 ,
4
1 6
)=
(z + 9z12 z22 + 8z32 + 6z2 z4 ),
2
24 1
which is obtained from 2.1.2. It yields the generating function
4
| y0 + y1 ) = y06 + y05 y1 + 2y04 y12 + 3y03 y13 + 2y02 y14 + y0 y15 + y16 ,
C(S4 ,
2
in accordance with figure 3.2. In order to make life easier we can replace y0 +. . .+ym−1 by 1+y1 +. . .+ym−1 ,
and in the case when |Y | = 2 we can even take 1 + y instead of y0 + y1 or 1 + y1 , so that, for example, the
generating function for the graphs on 4 vertices by their number of edges takes the following form:
4
| 1 + y) = 1 + y + 2y 2 + 3y 3 + 2y 4 + y 5 + y 6 .
C(S4 ,
2
Here the coefficient of y e is equal to the number of graphs on 4 vertices which possess exactly e edges, so
that we obtain (recall 2.1.3 and ??):
3.2.7 Corollary The total number of graphs on v vertices is equal to
v
C(Sv ,
; 2, . . . , 2),
2
22
while the number of selfcomplementary graphs on v vertices is equal to
v
; 0, 2, 0, 2, . . .).
C(Sv ,
2
The number of graphs on v vertices which contain e edges is the coefficient of y e in the polynomial
v
C(Sv ,
| 1 + y).
2
These examples have shown that the evaluation of the cycle indicator polynomial is an important step
towards the solution of various enumeration problems, so that a few remarks concerning this are well in
order. The cycle indicators of the natural actions of Sn , An , Cn , and Dn are at hand and we therefore
put the question how we can obtain cycle indicators of groups, which are direct sums or certain products of
symmetric, alternating, cyclic or dihedral groups. More generally we want to know how the cycle indicator
of a sum or a product can be obtained from the cycle indicators of the summands or factors, depending
of course on the action in question. To write obtained formulas in a compact form we introduce (recall)
the so-called cross-product (or combinatorial product)1 of polynomials, which is determined by the following
multiplication of monomials
Y
Y
Y Y
bj
b
ai
cr
3.2.8
ui ×
ui
=
uai i × ujj ,
i∈m
1 Palmer
j∈n
and Robinson call this the cartesian product.
i∈m j∈n
92
Chapter 3. Weights
where
b
a b gcd(i,j)
i j
uai i × ujj = ulcm
(i,j)
.
We shall write a×k = a × . . . × a (k times) to distinguish it from ak and
Xi∈n ai for a1 × . . . × an .
The following first remarks are clear from the definitions 1.2.8 and 1.2.10 of direct sum and cartesian product:
cycsumprod
3.2.9 Lemma Let
GX
and
HY
denote finite actions. We have
˙ ) = C(G, X) · C(H, Y ),
C(G × H, X ∪Y
and
C(G × H, X × Y ) = C(G, X) × C(H, Y ).
Further important constructions are the plethysm H G which we introduced in 1.5.2, and the composition
which was introduced in 1.5.8. They are defined as permutation groups induced by similar actions of H oX G,
and so the corresponding cycle indicator polynomials are the same:
same
compcyc
3.2.10
C(G[H], Y × X) = C(H G, |Y ||X|).
From the description of the conjugacy classes of elements of complete monomial groups given in 1.5.7 we
obtain for C(H G, |Y ||X|) the following plethysm of cycle indicators (recall 1.5.7, in particular its last
item):

ak (ḡ)
|X|
|Y |
Y
X
X
Y
1
a (h̄)
 1
3.2.11
.
zi·ki 
C(H, Y ) C(G, X) :=
|G| g
|H|
i=1
k=1
h
The last formula gives the cycle indicator for the exponentiation group [H]G of H by G (see 1.3.6)
expocyc
3.2.12

×ak (ḡ)
|k
X |X|
X |Y
Y
1
1

C([H]G , Y X ) :=
zrαr (k,h) 
,
|G| g k=1 |H|
r=1
X
h
where
expocyc.al
3.2.13
αr (k, h) =
X
1X
µ(s, r)
r
s|r
s
l
gcd(k,s)
.
lal (h̄)
gcd(k,s)
The Moebius function which is used in the above formula is determined by the partial order in N∗ , the set
of positive natural numbers. This order is defined by divisibility, i.e. p ≤ q ⇔ p|q. The corresponding µ is
called the number theoretic Moebius function.
We say that two posets (P, ≤) and (P 0 , ≤0 ) are order isomorphic if there exis a bijection Φ: P → P 0 and for all
p, q ∈ P we have p ≤ q ⇔ Φ(p) ≤0 Φ(q) (or, using the zeta function, we can write that ζ(p, q) = ζ(Φ(p), Φ(q)).
It means that for order isomorphic posets the corresponding zeta functions and, due to the recursions 2.4.4,
the Moebius function are identical since they are defined by the partial order. Therefore, considering (N∗ , |)
one can write µ(q/p) := µ(1, q/p) instead of µ(p, q) since the intervals [p, q] and [r, s] are order isomorphic if
q/p = s/r, and so µ(p, q) = µ(r, s).
In order to apply 3.2.12, we need to know the values of the number theoretic Moebius function µ(n) for
n ∈ N∗ . These values can be calculated by noting that the following is true:
numtheor
3.2.14 Corollary Let n has the prime number decomposition n = pk11 · . . . · pkr r , where the pi are different
prime numbers and the ki ≥ 1. Thus, the interval [1, n] is order isomorphic with
[1, pk11 ] × . . . × [1, pkr r ]
and
µ(n) = µ(pk11 , . . . , pkr r ) =
r
Y
i=1
µ(pki i ).
3.2. Cycle indicator polynomials
93
αr
1
Proof: For d ∈ [1, n] we have d|n, so d can be written as pα
1 · . . . · pr with 0 ≤ αi ≤ ki . It is evident that
the following mapping
αr
1
d 7→ (pα
1 , . . . , pr )
is a bijection wich determines an order isomorphism. The second equality is a consquence of the following
lemma (the proof is left as exercise 3.2.7):
3.2.15 Lemma From finite posets P, Q and their Moebius functions µP , µQ , we obtain the Moebius function
µP ×Q of P × Q, where
(p1 , q1 ) ≤ (p2 , q2 ) ⇐⇒ p1 ≤ p2 , and q1 ≤ q2 ,
in the following way:
µP ×Q ((p1 , q1 ), (p2 , q2 )) = µP (p1 , p2 ) · µQ (q1 , q2 ).
2
But from 2.4.4 we can deduce that µ(1) = 1, µ(p) = −1, if p is prime, and µ(pr ) = 0, if r > 1. This together
with 3.2.14 finally yields:
(
1
if n = 1,
nu
3.2.16
µ(n) = (−1)r if n is a product of r different primes,
0
otherwise.
Using the number theoretic Moebius function we can express some properties of the cyclic groups in a more
explicite way. For example, we consider the orbits of maximal length of Cn on mn (cf. 2.4.14). The numbers
lmn := |Cn \\n mn | =
3.2.17
1X
µ(d)mn/d
n
D
d|n
are called Dedekind numbers. Some of the smaller ones are shown in the following table:
1
1
2
3
4
5
6
7
m\n
1
2
3
4
5
6
7
2
0
1
3
6
10
15
21
3
0
2
8
20
40
70
112
4
0
3
18
60
150
315
588
5
6
7
0
0
0
6
9
18
48
116
312
204
670
2340
624
2580 11160
1554
7735 39990
3360 19544 117648
Finally we show how the cycle indicator polynomial C(G, X) can be obtained from the character χ: g 7→ |Xg |
of the action G X in question. In order to do this we have to evaluate the numbers ai (ḡ), i ∈ N, for each
g ∈ G, and we shall do this by inverting the equations
3.2.18
χ(g k ) = a1 (g¯k ) = a1 (ḡ k ) =
X
d · ad (ḡ)
ch
d|k
(cf. 1.4.10 for the last equation). By an application of 2.4.3 we obtain:
3.2.19 Corollary For each finite action
∀ k : a1 (ḡ k ) =
X
GX
the following equivalent systems of equations hold:
d · ad (ḡ) and ∀ k : ak (ḡ) =
d|k
1X
µ(k/d)a1 (ḡ d ),
k
d|k
where µ denotes the number theoretic Moebius function. In particular we obtain the following expression of
the cycle indicator of G X in terms of the character χ of G X:
C(G, X) =
1 X Y i−1 Σd|i µ(i/d)χ(gd )
z
.
|G| g i i
M
94
Chapter 3. Weights
3.2.20 Examples
ex
• The exterior cycle index of Ḡ ≤ SX is defined to be the cycle indicator of the natural action of SX on
the set of left cosets SX /Ḡ. The permutation character of this action is (exercise 3.2.11)
χ(π) =
|X|!|C S (π) ∩ Ḡ|
.
|Ḡ||C S (π)|
This together with an application of 3.2.19 yields the exterior cycle index
C(SX , SX /Ḡ)
of Ḡ.
• The character of SX on the set
X
k
of all the k–subsets of X is
k XY
ai (π)
χ(π) =
,
bi
i=1
b`ak
since a k–subset is fixed under π if and only if it is the union
Hence we
of cyclically permuted points.
v
X
,
can apply 3.2.19 in order to evaluate C(G, X
)
=
C(
Ḡ,
).
A
particular
case
is
C(S
v 2 ) which we
k
k
need for the enumeration of graphs on v vertices, and which is obtained from 2.1.2:
a1 (π)
χ(π̄) =
+ a2 (π).
2
Since Sv acts transitively on v2 then from 1.1.14 and 1.4.6 we have:
X a1 (π)
+ a2 (π) = v!,
2
π∈Sv
X
π∈C
a1 (π)
+ a2 (π)
Q 2 a (π)
= 1.
i
i
ai (π)!
i
3
Exercises
E221w
E 3.2.1
Check the equation 3.2.6.
E 3.2.2 Determine the cycle indicator polynomial C(D4 , 42 ) and next find the number of graphs on 4
vertices of the square and with e edges with respect to the action of the dihedral group D4 .
E221v
E 3.2.3 Evaluate the cycle indicator polynomial of the action of the group Cp × Cp × Cp , p being an
odd prime, on itself by left multiplication. Evaluate also the cycle indicator polynomial of the action of the
nonabelian group of order p3 on itself by left multiplication.
E221
E 3.2.4
Express the cycle indicator C(Sn , n) in terms of the polynomials C(Sk , k), 1 ≤ k ≤ n.
comcycsub
E 3.2.5
Let Ck (H, Y ) = C(H, Y ; zk , z2k , . . . , z|Y |k ). Show that
C(G[H], Y × X) = C(G, X; C1 (H, Y ), C2 (H, Y ), . . . , Cn (H, Y )).
hypocta
E 3.2.6 Using formula 3.2.12 determine the cycle indicators for S2 o S2 ' D4 and S2 o S3 ' Oh and compare
the obtained results with the previously used formulas.
3.2. Cycle indicator polynomials
95
Emulti
E 3.2.7
Prove 3.2.15.
InvIncExP
E 3.2.8 Determine the Moebius function for the lattice of all subsets of a finite set X (n, for example).
Using the Moebius Inversion 2.4.3 express the number of elements which have all properties in terms of
numbers of elements which have not some of these properties (cf. 2.2.7 and 2.2.8).
E222
E 3.2.9
Verify 3.2.16.
E 3.2.10
Show that
E
φ(n) =
X
d|n
d · µ(n/d), and n =
X
φ(d).
d|n
E 3.2.11
Prove 3.2.20.
E 3.2.12
Evaluate the characters of the natural actions of SX on X n and on Xi .
file:f32.tex
E
n
E
96
3.3
Chapter 3. Weights
Sums of cycle indicators, recursive methods
If AX denotes the alternating
group on the finite set X then, by 3.1.8, the coefficient of the monomial
P
Πy y c(f,y) in C(AX , X |
y) is equal to the number of AX –classes of mappings with the same content as
f . If we exclude the trivial case |X| = 1, then the AX –class of f differs from its SX –class if and only if
its stabilizers (SX )f and (AX )f are equal, and if these classes differ, then the SX –class of f consists of two
AX –classes.
The stabilizer of f in AX is the intersection of (SX )f given by 2.4.10 with AX . As each noninjective f is
left fixed by a transposition while each injective f has the identity subgroup as its stabilizer, we obtain
3.3.1 Corollary The stabilizers of f ∈ Y X in SX and in AX are the same, or, equivalently, they both are
equal to the identity subgroup, if and only if f is injective.
Thus the difference of cycle indicators of alternating and symmetric groups has the following useful interpretation (recall figure 1.4):
3.3.2 Corollary For |X| > 1 the polynomial
C(AX , X |
X
y) − C(SX , X |
X
y)
is the generating function for the enumeration of the injective SX –classes on Y X by weight.
Corresponding results hold for cyclic and dihedral groups. Also these arguments can be considered as a
certain kind of involution principle. In fact we obtain in the same way the following generalization of 3.3.2
(recall 1.4.21):
3.3.3 Corollary If
GX
denotes a chiral action, then
X
X
C(G+ , X |
y) − C(G, X |
y)
is the generating function for the enumeration by weight of the G–classes on Y X which split over G+ .
Besides these differences of cycle indicator polynomials we can form sums of cycle indicator polynomials of
series of groups, e.g. we obtain (by simply comparing coefficients) the equation
3.3.4
∞
X
n=0
C(Sn , n) = exp
∞
X
zk
k=1
k
∈ Q[[z1 , z2 , . . .]],
if C(S0 , 0 | p(u)) := 1 and where Q[[z1 , z2 , . . .]] denotes the ring of formal power series over Q in the indeterminates z1 , z2 , . . .. An immediate consequence is
3.3.5
∞
X
n=0
C(Sn , n | p(u)) = exp
∞
X
1
p(uk ) ∈ Q[[u]],
k
k=1
This sum is of great interest since it can be used for a recursion method which we are going to describe next.
3.3.6 Example A graph is called a tree if and only if it is connected and does not contain any cycle. A tree
with a single distinguished vertex is called a rooted tree. Figure 3.3 shows the smallest rooted trees. The
root is distinguished by indicating it as a circle while the other vertices are indicated by a bullet.
If r(x) denotes the generating function for the enumeration of rooted trees by their number of vertices, then
figure 3.3 shows that
r(x) = x + x2 + 2x3 + 4x4 + . . . .
We claim that this formal power series satisfies a recursion formula in terms of cycle indicator polynomials
of symmetric groups.
3.3. Sums of cycle indicators, recursive methods
3.3.7 Lemma r(x) = x
P∞
n=0
97
C(Sn , n | r(x)).
Proof: It is clear that the generating function for the enumeration of rooted trees is equal to the sum over
all n ∈ N of the generating functions of rooted trees with root degree n, i.e. where the root is incident with
exactly n edges. It therefore remains to evaluate the generating function for the enumeration of trees with
root degree n.
Denote by R the set of all the rooted trees. A rooted tree with root degree n can be considered as an orbit
of Sn on the set Rn . If we now give an element of R the weight xv , where v denotes the number of vertices
of the rooted tree in question, then we obtain the power series xC(Sn , n | r(x)) which therefore is the desired
generating function for the enumeration of rooted trees with root degree n by their number of vertices. This
completes the proof.
2
An application of the above exponential expressions 3.3.4 and 3.3.5 for the sum of the cycle indicator
polynomials of finite symmetric groups as an exponential formal power series allows to rewrite the recursion
formula for the generating function of the rooted trees in the following way:
3.3.8 Corollary The formal power series r(x) which generates the numbers of rooted trees by number of
vertices satisfies the recursion formula
r(x) = x · exp
∞
X
r(xk )
1
k
.
This recursion together with a suitable program system, like MAPLE or MACSYMA, allows to evaluate the
smallest numbers of rooted trees, some of which are shown in the following table:
n
1
2
3
4
5
6
7
8
9
10
bn
1
1
2
4
9
20
48
115
286
719
n
11
12
13
14
15
16
17
18
19
20
bn
1842
4766
12486
32973
87811
235381
634847
1721159
4688676
12826228
3
Similar arguments apply in each case when the structure in question consists of a certain and well defined
number of structures of the same kind.
3.3.9 Example Each graph is a disjoint union of connected graphs (a graph is called connected if and only
if from each of its vertices we can reach any other vertex by walking along suitably chosen edges). These
connected subgraphs are well defined, and they are called the connected components. Thus the generating
r
r
c
r
r
c
c
r
r
r
A Ac
r
c
r
A
Ar
r
A Ac
r
r
A Ar
c
r r r
A Ac
Figure 3.3: The smallest rooted trees
98
Chapter 3. Weights
P
P
function g(x) = v gv xv of the graphs and the generating function c(x) = v cv xv for the connected graphs
are related as follows:
∞
∞
X
X
c(xk )
3.3.10
g(x) =
.
C(Sn , n | c(x)) = exp
k
n=0
k=1
As we already know that
gv =
1 X c(π̄)
2
,
v!
π∈Sv
we can use 3.3.10 to evaluate the entries of the following table:
v
1
2
3
4
5
gv
1
2
4
11
34
cv
1
1
2
6
21
v
6
7
8
9
10
gv
156
1044
12346
274668
12005168
cv
112
853
11117
261080
11716571
3
Further examples show up in the sciences. It was already briefly mentioned that the origins of this theory
of enumeration lie in chemistry and the problem of isomerism. A few remarks concerning the history are
therefore in order. It was already in the 18th century when Alexander von Humboldt conjectured that there
might be chemical substances which are composed by the same set of atoms but have differend properties.
In his book with the title “Versuche über die gereizte Muskel- und Nervenfaser, nebst Vermutungen über
den chemischen Prozeß in der Tier- und Pflanzenwelt”, published in 1797, he writes (on page 128 of volume
I):
Drei Körper a, b und c können aus gleichen Quantitäten Sauerstoff, Wasserstoff, Kohlenstoff,
Stickstoff und Metall zusammengesetzt und in ihrer Natur doch unendlich verschieden seyn.
But it needed a quarter of a century to develop the anaytical methods that allowed to find out what the
atomic constituents of a chemical molecule are. These methods were developed in particular by J. L. Gay–
Lussac and J. von Liebig, who were the first to prove von Humboldt’s conjecture to be true. Here is a
sentence taken from a paper of Gay–Lussac that describes their discovery:
comme ces deux acides son très différents, il faudrait pour expliquer leur différence admettre
entre leurs éléments un mode de combinaison différent. C’est un objet qui appelle un nouveau
examen.
It needed some time to realize that a new phenomenon was discovered, J. J. Berzelius gave it the name
isomerism. Chemists tried to find out what the reason is by sketching molecules. Here are three of these
attempts to draw the molecule of C2 H5 OH: The first attempt is due to Couper:
O · · · OH
C
H2
..
.
C
· · · H3
The next one is the way how Loschmidt drew this molecule:
h h
h
lh
h h The breakthrough towards a solution of this problem is due to Alexander Crum Brown, who used a variation
of Loschmidt’s method by putting the circles of atoms that are assumed to be connected somehow apart but
joining them by edges in order to emphasize the connection. Here are his drawings for the alcohol C3 H7 OH:
3.3. Sums of cycle indicators, recursive methods
Hg
Hg
Hg
Hg
Cg
Cg
Cg
Hg
Hg
Hg
99
Og
Hg
Hg
Hg
Hg
Hg
Cg
Cg
Cg
Hg
Og
Hg
Hg
Hg
This introduction of molecular graphs solved the problem of isomerism by showing that there may exist
different graphs that correspond to a given chemical formula (which prescribes in fact the degree sequence,
which is the sequence of valencies of the atoms in the given formula). Moreover it gave rise to the development
of graph theory (there are of course also other birthdays of graph theory known, for example Euler’s solution
of the Königsberg bridge problem, and Kirchhoff’s invention of electric circuits as well as Hamiltons game
called “trip around the world”), and it stimulated the combinatorial theory of enumeration since the question
of chemical isomerism is at first glance equivalent to the problem of constructing all the graphs which are
connected and which have a given edge degree sequence. Let us consider, for example, the chemical formula
C3 H7 OH again, or, more generally the formula Cn H2n+1 OH. The problem is to construct all the graphs
which correspond to each of these formulae, i.e. all the graphs that consist of n vertices of degree 4 (the
carbon atoms are of that valency) together with 2n + 2 vertices of degree 1 (the hydrogen) and one of degree
2 (the oxygen) which must have a neighbour of degree 1 (so that a hydroxyl group −OH occurs). This
problem amounts to the construction of rooted trees, as, by a well–known result of graph theory, such degree
sequences can be satisfied by trees only (the cyclomatic number is zero in this case). The root represents
the substructure ≡ C − O − H, so that the root degree is ≤ 3. Consider the two examples shown above
which correspond to the formula C3 H7 OH. As the carbon atoms are of valency 4, we can make life easier
by neglecting the hydrogen atoms, so that the skeletons remain, which correspond to the rooted trees with
3 vertices. From each one of these structures we can reconstruct the original molecular graph and hence the
desired generating function a(x) for the numbers of alcohols satisfies the recursion
3.3.11
a(x) = x
3
X
C(Sn , n | a(x)).
n=0
The presented examples show that the cycle indicator has many applications and it is a very powerful tool
in enumeration problems. In the next section we shall count orbits by type and by weight what will serve
as another important example. Below we provide you with the cycle indicator of the symmetry group of a
hypercubic lattice, i.e. of the exponentiation [DN ]Sn (cf. 1.6.13).
To begin with we note that a set of mappings f : S → R, where S is any set and R is a ring of polynomials
Q[z1 , z2 , . . .], can be endowed with ring structure introducing
scalar multiplication over Q:
(λf )(P ) := λ · f (P ),
pointwise addition of mappings: (f + g)(P ) := f (P ) + g(P ),
cross-product of mappings:
(f × g)(P ) := f (P ) × g(P ),
where λ ∈ Q, f, g ∈ RS and P ∈ S. We are interested in the case when S = R and these mappings are
Q-linear. It means, in particular, that for the cycle indicator C(H, Y ; z1 , . . . , z|Y | ) we have
|Y |
1 X Y aj (h̄) f
zj
,
f C(H, Y ) =
|H|
j=1
h∈H
i.e. f is determined by its values at monomials. (Recall that each monomial in the above sum describes the
cycle type of h̄ ∈ H̄.) For each k ∈ N∗ we introduce a Q-linear mapping f ∈ RR such that
3.3.12
fk
|Y |
Y
j=1
|Y |k
a
zj j
=
Y
r=1
where αr (k) are given by 3.2.13. These definitions lead to
zrαr (k) ,
m
100
Chapter 3. Weights
3.3.13 Theorem (Palmer and Robinson) The cycle indicator C([H]G , Y X ) is the image of C(H, Y ) P
under the mapping obtained by substituting the operators fk for the variables uk in the cycle indicator
C(G, X; u1 , . . . , u|X| ) of G X; symbolically
C([H]G , Y X ) = C(G, X; f1 , . . . , f|X| ) C(H, Y ) .
Before we apply this theorem to [DN ]Sn let us consider this formula in more details taking into account
results of section 1.5. Let pi ∈ R be a monomial determined by the cycle type of a(h̄) of elements in the i-th
class of H. Then
|Y |
X |C i |
X |C i | Y
a (h̄)
=
C(H, Y ) =
zj j
pi ,
|H| j=1
|H|
i∈m
i∈m
where C 1 , C 2 , . . . , C m are classes of H. The cycle indicator of Sn acting on n is given in example 3.2.3 so
we obtain from 3.3.13
X |C i |
×ak
X
(ak !k ak )−1
C([H]Sn , |Y |n ) =
fk (pi )
.
|H|
k∈n
i∈m
aàn
X
The Multinomial Theorem
(x1 + x2 + . . . + xm )n =
X
(n)
where the sum is taken over all m-tuples (n) with
P
Y
n!
xni ,
n1 ! · · · nm ! i∈m i
ni = n, yields that
×ak X
×aik
X |C i |
|C i |
ak !
fk (pi )
=
fk (pi )
.
|H|
a1k ! · · · amk ! i∈m |H|
i∈m
X
(ak )
Let C a denote a class of the complete monomial group H o Sn of the type a = (aik ) (cf. 1.5.6 and 1.5.7).
Indentifying (ak ) with the k-th column of this matrix we finally obtain
excyc.type
cocyc.type
3.3.14
C([H]Sn , |Y |n ) =
×aik
1 X a
|C
|
f
(p
)
.
k
i
|H|n n! a
k∈n i∈m
XX
Taking into account formula 3.2.11 and exercise 3.2.5 we can write the cycle indicator for a composition
Sn [H] in the similar form as
1 X a YY
C(Sn [H], n|Y |) =
(pi,k )aik ,
3.3.15
|C |
|H|n n! a
i∈m
k∈n
where
pi,k := pi (zk , z2k , . . . , z|Y |k ).
It follows from this formulas and from equation for the order of elements in the class C a (see **) that the
upper limit of the product in 3.3.12 can be taken as lcm j:aj 6=0 (j)k. The formal proof is too cubersome but
we will see that it is correct in the example which follows.
The dihedral group DN , which we are going to consider, acting on N leads to the cycle indicator with all
pi being powers of a single indeterminate zd with d|N or products z1a1 z2a2 , where a1 and a2 depend on the
N/d
parity of N (see 3.2.3). Therefore, we need to consider only images fk (zd ) and fk (z1a1 z2a2 ) for k ∈ n. The
first immediate result is that
f1 (pi ) = pi
for any pi (not only for those mentioned above). Let us recall the definition 3.2.13 of αr :
αr (k, h) =
X
1X
µ(r/s)
r
s|r
s
l
gcd(k,s)
gcd(k,s)
lal (h̄)
.
3.3. Sums of cycle indicators, recursive methods
N/d
If pi = zd
101
then l has to be equal d and henceforth
s
=d
gcd(k, s)
since we have assumed that r (so also s) is not greater than dk. It can be proved (and we again omit this
proof but the obtained results will show the correctness of our presumptions) that r has to be of the following
form
r = dκ gcd(k, d),
where κ divides k/ gcd(k, d) = lcm (k, d)/d, so gcd(κ, d) = 1. The same is valid for s, but since s|r then we
have
s = dκ0 gcd(k, d), where κ0 |κ,
and s/ gcd(k, s) = d for all κ, κ0 . Finally, for h̄ = (1 . . . N )N/d we have
0
1X
µ(κ/κ0 )N κ gcd(k,d) .
d gcd(k, d)αr (k, h) =
κ 0
κ |κ
0
0
0
Replacing κ by the complementary divisor κ̄ = κ/κ we obtain
0
1X
d gcd(k, d)αr (k, h) =
µ(κ̄0 )(N gcd(k,d) )κ/κ̄ = lηκ ,
κ 0
κ̄ |κ
where η = N
Therefore,
gcd(k,d)
and lηκ is the number of orbits of length κ in the action of Cκ on η κ (cf. 3.2.17).
N/d
fk (zd
Y
)=
κ
l
/d gcd(k,d)
ηκ
zdκ
gcd(k,d)
.
k
gcd(k,d)
Note that these formulas can be used to determination of the cycle indicator for the complete monomial
group CN o Sn . If k = n then the cycle type a à n has to be (0, . . . , 0, an = 1) and ain = 1 for one i only
and therefore the above formula provides us with the final form of monomials corresponding to these classes.
Since the subscript is the length of a cycle and the superscript is the number of cycles then should be equal to
the order of a set the considered group acts on. In the considered case it follows from ithe Moebius Inversion
2.4.3 and 3.2.17 that
X
κ · lN gcd(n,d) ,κ = N n .
n
κ
gcd(n,d)
In the considerations of the other classes it may be necessary to raise the obtained formula to the m-th
cross-product power. It is not difficult to show that
Y Y gcd(κi )lηκ ···lηκ /d gcd(k,d)
N/d ×m
m
1
fk (zd )
=
···
zd gcd(k,d)lcm
,
(κi )
κ1
κm
k
for j ∈ m. We stop2 our general consideration here and we find the cycle indicators for the
where κj | gcd(k,d)
exponentiations [DN ]S2 and [DN ]S3 . The corresponding polynomials in indeterminates uk , which are to be
replaced by fk , are as follows
1
C(S2 , 2) = (u21 + u2 )
2
and
1
C(S3 , 3) = (u31 + 3u1 u2 + 2u3 ).
6
(Note that the coefficients of these monomials have been taken into account in 3.3.14 since they correspond
to class orders.) Since k = 1, 2 or 3 and none of ak > 1 for k > 1, so we do not need the last formula (recall
that f1 (pi ) = pi ). In both cases we have to consider separately k|d and k - d and we obtain
 N/d (N 2 −N )/2d

zd z2d
k = 2, d odd;



2

N
/2d
z
k = 2, d even;
N/d
2d
fk (zd ) =
N/d (N 3 −N )/3d


zd z3d
k = 3, 3 - d;



 N 3 /3d
z3d
k = 3, 3|d.
2I
think that we can stop also somewhere earlier since these formulas become too much complicated. WSF
102
Chapter 3. Weights
Now we must consider fk (z1a1 z2a2 ). According to our presumption it is enough to find αr for r ≤ 4 and r ≤ 6
for k = 1 and 2, respectively. In the first case (k = 2) we obtain;
α1
α2
α3
α4
= a1 ;
= (a21 − a1 )/2;
= 0;
= (a1 + a2 )a2 .
Hence
(a21 −a1 )/2 (a1 +a2 )a2
z4
.
f2 (z1a1 z2a2 ) = z1a1 z2
If z1a1 z2a2 describes a cycle structructure of h̄ acting on an N -element set then it has to be N = a1 + 2a2 .
The obtained monomial corresponds to an action on a set consisting of
1 · a1 + 2 · (a21 − a1 )/2 + 4 · (a1 + a2 )a2 = a21 + 4a1 a2 + 4a22 = N 2
elements what agrees with the order of N n in the considered case. In the similar way we obtain
(a31 −a1 )/3 (3a21 a2 +6a1 a22 +4a32 −a2 )/3
z6
.
f3 (z1a1 z2a2 ) = z1a1 z2a2 z3
It is easy to determine the other necessary formulas
N/d ×m
(zd
)
(z1a1 z2a2 )×m
file:f33.tex
N m /d
= zd
am
1
;
[(a1 +2a2 )m −am
1 ]/2
= z1 z2
=
;
3.4. Weighted enumeration by stabilizer class
3.4
103
Weighted enumeration by stabilizer class
In section 2.4 we considered counting of orbits by type, i.e. by conjugate class of stabilizer. This classification
can also be presented in the weighted form providing us with detailed description of orbit properties.
The proof of the weighted form 3.1.1 of the Cauchy-Frobenius Lemma was as easy as the proof of its constant
form 1.1.12; the same holds for the corresponding weighted form of Burnside’s Lemma, which we are going
to introduce next. Recall that, by Burnside’s result 2.4.2, we obtain for the elements t that represent orbits
e in a transversal T of all the orbits of G on X:
of type U
e \\\X| =
|U
X
1=
X
e| X
|U
1.
µ(U, V )
|G/U |
V ≤G
e
t∈T :Gt ∈U
x∈XV
If we now replace the 1’s on both sides by the weight of the element to which they correspond, then we
obtain the identity
X
X
e| X
|U
w(t) =
w(x).
µ(U, V )
|G/U |
x∈X
V
≤G
V
e
t∈T :Gt ∈U
Since the bijection described in 2.4.5 is weight preserving, we get
X
3.4.1
w(t) =
ei
t∈T :Gt ∈U
re
X
ei | X
|U
ek )
µ(Ui , U
w(x).
|G/Ui |
x∈XUk
k
A direct consequence is the desired result on the enumeration by weight and stabilizer class:
3.4.2 Burnside’s Lemma, weighted form Let G X denote a finite action and w: X → R a function from 33
X into a commutative ring R which contains Q as a subring. If w is constant on the orbits of X, then we
have, for the elementes t of a transversal T of the orbits and the vector of the sums of weights of transversals
ei \\\X of G on X, the equation
of strata U

..
.


..
.

P

P


w(t)  = B(G) ·  x:Ui ≤Gx w(x)  .
 t:Gt ∈Uei



..
..
.
.
This weighted form of Burnside’s Lemma was, as far as I know, first stated and proved in P. Stockmeyer’s
thesis. He provided applications of the following immediate consequence to the enumeration of graphs by
weight and type:
ej by weight 33
3.4.3 Corollary
The generating function for the enumeration of G–classes on Y X of type U
Q
w: f 7→ f (x) ∈ Q[Y ] is the j-th row of the following one column matrix:



..
..
.
 Q|Ui \\X| P

 Q|X| . ak (Ui ) 
lν (Ui ) 


y
B(G) · 
=
B(G)
·
y
 ν=1

 k=1 zk

..
..
.
.

,
zj :=Σy j
where lν (Ui ) denotes the length of the ν-th orbit of Ui on X and ak (Ui ) = |Ui \\k X| is the number of orbits
of the length k in Ui \\X.
3.4.4 Example For the enumeration by type and weight of the classes of spin configurations with respect by
104
Chapter 3. Weights
to the dihedral group, which we already considered in example 2.4.9, we obtain that the generating functions
are the rows of the following matrix:

 4   4
(z1 − z22 − 2z12 z2 + 2z4 )/8
z1

 z22  
0

 2  
2

 z2  
(z2 − z4 )/2

 2  
2
2

 z1 z 2  
(z1 z2 − z2 )/2
.
=
B(D4 ) · 

 z4  
0

 2  
2

 z2  
(z2 − z4 )/2

 


 z4  
0
z4
z4
Let us, for example, refine the third row in table 2.1, i.e. consider s = 1. We can do it for the most general
weight substituting zj = y1̄j + y0j + y1 or for the magnetization only replacing zj by u0 + uj + u2j (recall that
in the latter case a monomial qum means that there are q orbits with magnetization M = m − ns = m − 4).
These substitutions yield the following vectors:


y1̄2 y0 y1 +y1̄ y02 y1 +y1̄ y0 y12


0


2 2
2 2
2 2


y1̄ y0 +y1̄ y1 +y0 y1

 3
 y y0 +y 3 y1 +y1̄ y03 +y1̄ y13 +y03 y1 +y0 y13 +y 2 y0 y1 +y1̄ y02 y1 +y1̄ y0 y12 
1̄
1̄

 1̄


0


2 2
2 2
2 2


y1̄ y0 +y1̄ y1 +y0 y1




0
4
4
4
y1̄ +y0 +y1
and












u3 + u4 + u5
0
u2 + u4 + u6
u + u2 + 2u3 + u4 + 2u5 + u6 + u7
0
u2 + u4 + u6
0
u0 + u4 + u8






.





These vectors show, for example, that each of 9 orbits of type U4 = S2 has different weight and orbits with
magnetization M = 0 (i.e. corresponding to u4 ) have all admissible types.
3
The first of above presented vectors can serve as a very good example of the Weyl duality for permutaions
(see 2.4.16). Each polynomial is invariant under the time-reversal symmetry, i.e. under the transposition
y1̄ ↔ y1 (in the second vector it corresponds to the transposition um ↔ u8−m ). Moreover, we see that each
e \\\Y X is an invariant
entry is invariant under any permutation σ ∈ S{1̄,0,1} . It verifies that each stratum U
subset under action of H ≤ S{1̄,0,1} and can be written as an union of the corresponding orbits. For the
sake of simplicity we replace S{1̄,0,1} by the isomorphic group S3 . Then the vector under discussion can be
written as


S3 (y1̄2 y0 y1 )


0


2 2


S3 (y1̄ y0 )


 S3 (y 3 y0 ) + S3 (y 2 y0 y1 ) 
1̄
1̄

,


0


2 2


S3 (y1̄ y0 )




0
4
S3 (y1̄ )
where
Y
yiai ) =
1
X Y
ai
yσi
|(S
)
|
m
f
i∈m
σ∈Sm i∈m
Q
ai
and (Sm )f denotes a stabilizer of a monomial f = i∈m yi . According to the Weyl duality we can at first
consider action of S3 on {1̄, 0, 1}D4 and next the action of D4 on each stratum. It is done in the following
Sm (
3.4. Weighted enumeration by stabilizer class
105
3.4.5 Example We have not got a formula similar to 2.4.8 describing classification by type H-configurations,
so we must apply the Burnside’s Lemma 2.4.7 in the general form. There are four classes of conjugate
subgroups in S3 (S1 , S2 , C3 , S3 ) and we consider the action of H = S3 on 81-element set {1̄, 0, 1}4 . It is
not difficult to determine the Burnside matrix B(S3 ) and the numbers of fixed points under each subgroup
of S3 and hence we obtain that the length of strata are given by the following vector

 
 

1/6 −1/2 −1/6 1/2
81
13


 

1
·
−1 

 ·  1  =  1 .





1/2 −1/2
0
0 
1
0
0
It can be checked that 13 orbits in the first stratum form 5 orbits under action of D4 :
Representative
S3 (1̄, 1̄, 0, 1)
S3 (1̄, 1̄, 0, 0)
S3 (1̄, 1̄, 1̄, 0)
S3 (1̄, 0, 1̄, 1)
S3 (1̄, 0, 1̄, 0)
Order
4
2
4
2
1
Stabilizer
S20
D20
S2
D2
D4
Note that we cannot compare weighted enumeration since the weights used in the previous example are not
constant on orbits of S3 .
3
I am not satisfied with the previous example. I thought about something different but the result is as you see.
The following example was commented out. I have uncommented it temporarily but it isn’t so important for the
main aim of this book, so it can be removed, I think.
A further example is the action of a finite group G on the set Y X := mG induced by the regular action G G
fi by weight:
of G on itself. According to 3.4.3 we obtain the generating function for the orbits of type U
|Uk \\G|
X
bik
Y X
ν=1
k
l (Uk )
yjν
.
j∈m
The orbits of Uk on G are the right cosets of Uk , hence |Uk \\G| = |G/Uk | and lν (Uk ) = |Uk |, so that we have
proved
rm
3.4.6 Corollary The coefficient of the monomial y1r1 . . . ym
in the polynomial
|G/Ui |

X
bik 
33
X
|U |
yj k 
j∈m
k
fi of G on the set mG which are of content (r1 , . . . , rm ).
is equal to the number of orbits of type U
Now we note that, for f ∈ mG , U is a subgroup of the stabilizer if and only if f is constant on the right
cosets of U in G. Hence the stabilizer Gf is just the maximal subgroup of G such that f is constant on its
right cosets. We consider the particular case m := 2, where mG can be identified with the subsets of G. We
derive from 3.4.6 the result:
|G|−r
3.4.7 Corollary The coefficient of y1r y2
|G/Ui |
in the polynomial
X
|G/Ui |
|U |
|U |
bik y1 k + y2 k
k
is equal to the number of subsets S ⊆ G of order r with the property that the maximal subgroup U ≤ G such
ei .
that S is a union of right cosets of U in G, belongs to U
33
106
Chapter 3. Weights
The end of a ‘temporary example’. A next one is fine.
A case of particular interest is the enumeration by weight of asymmetric classes, i.e. of orbits with trivial
stabilizer class e
1. The corresponding generating function is direct from 3.4.1:
prim1
3.4.8
X
w(x) =
t∈T :Gt =1
X µ(1, Uk )
|NG (Uk )|
k
X
w(x).
x:Uk ≤Gx
This series is called the asymmetry
series. For example, the asymmetry series of the action
P
obtained by Pólya-inserting
y into the asymmetry indicator
prim2
3.4.9
A(G, X) :=
G (Y
X
) can be
X µ(1, Uk ) Y |U \\ X|
z k i ,
|NG (Uk )| i i
k
where Uk \\i X denotes the set of orbits of length i of Uk on X.
aseries
3.4.10 Corollary The generating function for the enumeration of asymmetric G-classes on Y X by multiplicative weight is
X
y).
A(G, X |
In the case of the cyclic group the asymmetry indicator is
prim3
3.4.11
A(Cn , n) =
1X
n/d
µ(d)zd .
n
d|n
In the case of the symmetric group we obtain the following asymmetry indicator:
A(Sn , n) =
Y |V \\ n|
1 X
µ(1, V )
zi i .
n!
i
V ≤Sn
In this particular example we have a second expression for this series, since the asymmetric orbits consist of
the injective mappings. This yields the following equation:
A(Sn , n) = C(Sn , n) − C(An , n).
Comparing the coefficients of the monomial zn on both sides of this equation we get the following result on
the Moebius function on the subgroup lattice of the symmetric group:
prim4
3.4.12
X
µ(1, V ) = (−1)n−1 (n − 1)!.
V transitive
The problem with applications of Burnside’s Lemma to finite actions G X is that they need the Burnside
matrix B(G) of G which assumes a detailed knowledge of the subgroup lattice L(G), and this information
is not easy to obtain. For example in the case of the symmetric groups Sn , the maximal n for which we can
get B(Sn ) by using sophisticated program systems is n = 8 or, maybe, n = 9. Most of this information is
redundant if we are dealing with a single action G X since only a few of these subgroups occur as stabilizers
of elements x ∈ X. Let us discuss these subgroups of G and the corresponding set partitions of X in some
detail. Consider the set
L(G) := {U | U ≤ G},
of subgroups of G, ordered by inclusion: (L(G), ≤), together with the set
SP (X) := {p | p is a set partition of X}
of set partitions p = {p1 , . . .} of X, ordered by refinement:
p ≤ q : ⇐⇒ ∀ pi ∈ p ∃ qj ∈ q: pi ⊆ qj .
3.4. Weighted enumeration by stabilizer class
107
We introduce the following mappings between these two sets:
per: L(G) → SP (X): U 7→ U \\X, stab: SP (X) → L(G): p 7→ Gp ,
where Gp := {g ∈ G | ∀i: g[pi ] = pi } is the stabilizer of p. The partition per(U ) = U \\X is called the period
of U . These mappings are monotone:
U ≤ V ⇒ U \\X ≤ V \\X, p ≤ q ⇒ Gp ≤ Gq ,
and their compositions satisfy:
U ≤ (stab ◦ per)(U ), p ≥ (per ◦ stab)(p).
This means (cf. exercises 3.4.1 and 3.4.2) that per is a Galois function, that stab ◦ per is a closure operator
on L(G), and that per ◦ stab is a coclosure operator on SP (X). Correspondingly we have for the quotients,
i.e. the set L̄(G) of closed elements, and the set SP (X) of coclosed elements, respectively:
3.4.13
L̄(G) = stab[SP (X)], SP (X) = per[L(G)].
34
Moreover these quotients are isomorphic (as posets):
3.4.14
L̄(G) ' SP (X).
34
3.4.15 Example A very easy case is provided by the natural action of the symmetric group G := Sn , on e3
the set X := n. The stabilizer of a partition p ∈ SP (n) is obviously the direct sum ⊕i Spi of the symmetric
groups Spi on the blocks pi of p. The closed elements, which form the quotient L̄(Sn ), are these direct sums,
which are called the Young subgroups of Sn . On the other hand, each element of SP (n) is coclosed, and so
we have:
L̄(Sn ) ' SP (X).
In words: the lattice of Young subgroups of Sn is isomorphic to the lattice of partitions of n. This shows
that in order to examine the natural action of Sn on n, it suffices to consider the lattice SP (n) of partitions
or the lattice of Young subgroups.
3
Now we use these notions in order to introduce periods of mappings . The partition p(f ) of f ∈ Y X has as
its blocks the subsets on which f is constant:
p(f ) := {f −1 [{y}] | y ∈ f [X]}.
The coclosure of p(f ) is called the period of f :
per(f ) := (per ◦ stab)(p(f )) = per(Gp(f ) ) = Gp(f ) \\X.
Less formally: per(f ) is the coarsest partition of the form U \\X such that f is constant on each block. Now
recall the multiplicative weight function
Y
w: Y X → Q[Y ]: f 7→
f (x).
x∈X
We shall use it in order to define two interesting mappings on the set of periods of
P (G X) := SP (X) = {p ∈ SP (X) | p = (per ◦ stab)(p)}
= per[L(G)] = {U \\X | U ≤ G}.
These mappings are:
A: P (G X) → Q[Y ]: p 7→
X
f :per(f )=p
w(f ),
G X:
108
Chapter 3. Weights
and
B: P (G X) → Q[Y ]: p 7→
X
w(f ).
f :per(f )≤p
They satisy the following equation:
3.4.16
B(p) =
X
µ(q, p)A(q),
34
q
which means that for each period p of G on X the following is true:
344
X
3.4.17
X
w(f ) =
f :per(f )≤p
q∈P (G X)
µ(q, p)
X
w(f ).
f :per(f )=q
This improves the methods offered by the weighted form of Burnside’s Lemma, since on the right hand side
there are considerably fewer summands.
Exercises
E341
E 3.4.1
Let (M, ≤) and (N, ≤) denote partially ordered sets and consider a pair (α, β) of mappings
α: M → N and β: N → M . Then (α, β) is called a Galois connection if and only if α and β are antitone3 ,
and
(β ◦ α)(m) ≥ m, (α ◦ β)(n) ≥ n.
Prove that in this case the following is true:
• α ◦ β ◦ α = α, and β ◦ α ◦ β = β.
• α ◦ β and β ◦ α are closure operators:
– m ≤ (β ◦ α)(m), and n ≤ (α ◦ β)(n),
– m ≤ m0 ⇒ (β ◦ α)(m) ≤ (β ◦ α)(m0 ),
– n ≤ n0 ⇒ (α ◦ β)(n) ≤ (α ◦ β)(n0 ),
– (α ◦ β)2 = α ◦ β, and (β ◦ α)2 = β ◦ α.
• For the subsets of closed elements (i.e. the m, n with the property m = (β ◦ α)(m), n = (α ◦ β)(n)) we
have
M̄ = β[N ], N̄ = α[M ].
• M̄ and N̄ are antiisomorphic, with α and β as inverse mappings.
E342
E 3.4.2 Let (M, ≤) and (N, ≤) denote partially ordered sets. A mapping α: M → N is called a Galois
function if there exists a mapping β from N to M such that both these mappings are monotone, while
(β ◦ α)(m) ≥ m, and (α ◦ β)(n) ≤ n.
Prove that in this case the following is true:
• β ◦ α is a closure operator on M , while α ◦ β is a coclosure operator on n, which means:
n ≥ (α ◦ β)(n), n ≤ n0 ⇒ (α ◦ β)(n) ≤ (α ◦ β)(n0 ), (α ◦ β)2 = α ◦ β.
• α[M ] is the set of coclosed elements (i.e. the set of n such that n = (α ◦ β)(n)).
• The set M̄ of coclosed elements of M and the set N̄ of coclosed elements of N are isomorphic with α
and β as inverse mappings.
I found this example in the previous version. I think that it fits this section but I cann’t figure out in which way.
3.4. Weighted enumeration by stabilizer class
exdouble
109
3.4.18 Example (Classes of configurations and double cosets) Assume that we are given the set of
configurations Y X , a specific configuration f ∈ Y X , and that we want to put our hands on the set of all
the configuration classes that consist of mappings of the weight c(f, −). A group that is transitive on this
particular set of configurations of weight c(f, −) is the symmetric group SX , and so the above result 1.2.6
can be applied. If (SX )f denotes the stabilizer of f in SX , while G\\c(f,−) Y X denotes the set of configuration
classes, consisting of elements of this particular weight, then we obtain the bijection
ϕ: G\\c(f,−) Y X → Ḡ\SX /(SX )f ,
where Ḡ denotes the permutation group (a subgroup of SX ) induced by G on X.
This shows clearly that from a transversal of these double cosets we can obtain a transversal of the configuration classes of prescribed weight. For instance, these considerations allow us to find all spin configurations
with given total magnetization M when s = 1/2. For larger spin we have to consider some specfic configurations since the same M can be obtained for configurations with different contents (e.g. when s = 1 and
n = 2 then M = 0 is provided by configurations (0, 0) and (−1, +1)).
3
A concrete example will demonstrate how this can be used in the constructive theory of discrete structures:
3.4.19 Example Specific classes of configurations are the simple graphs on a set V of v vertices, say. As gr
it was mentioned above, they correspond to orbits of the group G := SV on the set
V
Y X := {0, 1}( 2 ) .
An element f of this set is a labelled graph with v vertices, and it has e edges if and only if it takes the
value 1 exactly e-times. Hence a particular labelled graph with e edges is the mapping
f := (0, . . . , 0, 1, . . . , 1).
| {z }
e times
Its stabilizer is (if 1, . . . ,
v
2
denote the numbers of the pairs of vertices)
(SX )f = S{1,...,(v)−e} ⊕ S{(v)−e+1,...,(v)} =: S((v)−e,e) ,
2
2
2
2
a so-called Young subgroup. Hence the preceding example shows that the set of simple graphs on v points
containing exactly e edges is in one-to-one-correspondence with the set of double cosets
S̄V \S(V ) /S((v)−e,e) .
2
2
Thus we can
obtain the simple graphs on v vertices from a transversal of these sets of double cosets, for
0 ≤ e ≤ v2 . This looks cumbersome, and it is, but it has been successfully used in order to get complete
lists of simple graphs of up to ? vertices. In order to do this efficiently, B. Schmalz recently invented the
so-called ladder game to evaluate the desired representatives by going up and down in a ladder of subgroups
and carefully watching the corresponding decompositions and fusions of double cosets. For exammple, in
the case of v = 4 the subgroup ladder looks as follows:
3
file:f34.tex
3 What
does it mean? WSF
110
Chapter 3. Weights
S(6,0)
S(5,1)
S(4,2)
S(3,3)
S(4,1,1)
S(3,1,2)
Figure 3.4: Ladder of subgroups for the evaluation of graphs on 4 vertices
3.5. The Burnside ring
3.5
111
The Burnside ring
We saw how symmetry classes of mappings can be counted by weight and stabilizer class using the Burnside
matrix which is the inverse of the table of marks. Moreover we met actions of groups on lattices (L, ∧, ∨),
and the corresponding subrings ZL,∧
and ZL,∨
of the semigroup rings ZL,∧ and ZL,∨ , where the matrices
G
G
∧
∧
∨
∨
A = (aik ) and A = (aik ) played a central role. We now build a bridge between these two topics by
L(G),∧
introducing the Burnside ring Ω(G) of G which can be embedded into ZG
, the matrix A∧ of which
is closely connected with the table of marks. In order to do this we indicate first how a complete set of
transitive G–sets can be obtained (exercise 3.5.1):
e1 , . . . , U
ed and represen3.5.1 Lemma Let G denote a finite group with its conjugacy classes of subgroups U
e
tatives Ui ∈ Ui . Then
• The sets G/Ui := {gUi | g ∈ G}, i ∈ d, consisting of the left cosets gUi of the subgroups Ui of G are
G–sets with respect to the operation g : xUi 7→ gxUi .
• These G–sets are transitive and pairwise not similar.
• {G/U1 , . . . , G/Ud } is a complete system of transitive but pairwise dissimilar G–sets.
Switching to the similarity classes
G/Ui := {X | X a G–set and
GX
≈
G (G/Ui )},
we obtain the complete system Ω := {G/U 1 , . . . , G/U d } of G–similarity classes of transitive G–sets. Consider
the free abelian group
ZΩ := {ψ | ψ : Ω → Z}, where (ψ + ψ 0 )(G/Ui ) := ψ(G/Ui ) + ψ 0 (G/Ui ).
We shall as usual display the ψ ∈ ZΩ as “formal sums”
ψ=
d
X
zi · G/Ui , where zi := ψ(G/Ui ) ∈ Z.
i=1
The addition of the two formal sums
G/Ui := 1Z · G/Ui and G/Uk := 1Z · G/Uk
in ZΩ can be interpreted as first taking the disjoint union of G/Ui and G/Uk and afterwards switching to
the G–similarity class X of the resulting G–set X:
˙
G/Ui + G/Uk = G/Ui ∪G/U
k.
This is clear from the fact that each G–set possesses a unique decomposition into orbits (but note that also
˙
in the case when i = k we have to form the disjoint union, e.g. |G/Ui ∪G/U
i | = 2|G/Ui |). Hence each G–set
X and its similarity class X can be identified in this way with a uniquely determined element of the subset
NΩ ⊆ ZΩ :
d
X
ei \\\X| · G/Ui .
3.5.2
X=
|U
i=1
Furthermore we can introduce on ZΩ a multiplication as linear extension of
X
X
G/Ui · G/Uj :=
bijk G/Uk , if G/Ui × G/Uj =
bijk G/Uk .
k
k
Soon we shall see that this in fact yields a well defined multiplication. The corresponding structure constant
bijk is equal to the number of orbits of G on G/Ui × G/Uj (with respect to the natural action g(x, y) :=
(gx, gy)) that belong to the class G/Uk .
112
Chapter 3. Weights
For example, the conjugacy classes of subgroups in G := S3 are represented by the subgroups
U1 := {1}, U2 := S2 , U3 := A3 , U4 := S3 ,
and so we obtain the following transversal of the similarity classes of transitive S3 –sets:
G/U1 = S3 /{1}, G/U2 = S3 /S2 , G/U3 = S3 /A3 , G/U4 = S3 /S3 ,
which are of order 6,3,2,1, respectively. It is easy to check that S3 acts, for example, transitively on G/U2 ×
G/U3 , so we obtain (already by checking cardinalities) that
G/U 2 · G/U 3 = G/U2 × G/U3 = G/U 1 .
L(G),∧
3.5.3 Theorem The following mapping defines an embedding of the ring (ZΩ , +, ·) into the ring ZG
L(G),∧
: G/Ui 7→ |NG (Ui ) : Ui |ui ,
ui :=
X
ZΩ ,→ ZG
:
where
L(G),∧
U ∈ ZG
.
ei
U ∈U
L(G),∧
Proof: The structure constants of ZΩ were denoted by bijk , while the structure constants of ZG
a∧
ijk introduced in 2.6.9. It therefore remains to prove the following equation:
bijk =
are the
|NG (Ui )/Ui ||NG (Uj )/Uj | ∧
aijk .
|NG (Uk )/Uk |
ek | = |G/NG (Uk )| we have
Since |U
bijk =
=
=
1
ek }|
|{(x, y) ∈ G/Ui × G/Uj | G(x,y) ∈ U
|G/Uk |
ek |
|U
|{(x, y) ∈ G/Ui × G/Uj | G(x,y) = Gx ∩ Gy = Uk }|
|G/Uk |
|Uk |
ei × U
ej | U ∩ V = Uk }| |NG (Ui )||NG (Uj )|
|{(U, V ) ∈ U
|NG (Uk )|
|Ui ||Uj |
=
|NG (Ui )/Ui ||NG (Uj )/Uj | ∧
aijk .
|NG (Uk )/Uk |
2
This shows in particular that
Ω(G) := (ZΩ , +, ·)
is a ring which we call the Burnside ring of G. The table of marks M (G) of G is closely related to the table
L(G),∧
A∧ = (a∧
:
ik ) of ZG
L(G),∧
3.5.4 Theorem The matrix A∧ of ZG
and the table M (G) of marks of G satisfy the equation


..
0 
 .
.
M (G) = A∧ · 
|N
(U
)/U
|
G
k
k


..
.
0
Proof: Using 2.5.1 and 2.6.6 we obtain:
mik =
|G/Uk |
ek | Ui ≤ V }| = |NG (Uk )/Uk |a∧ .
|{V ∈ U
ik
ek |
|U
2
L(G),∧
ZG
,
We have just seen how the Burnside ring Ω(G) can be embedded into
L(G),∧
from 2.6.10 that ZG
is isomorphic to Zd via ui 7→ a∧
i so that we finally obtain
and we already know
3.5. The Burnside ring
113
3.5.5 Corollary The following mapping linearly extends to an embedding of rings:

m1k
.
ε : Ω(G) ,→ Zd : G/Uk 7→ µk :=  ..  .
mdk

We call these columns µk of M (G) the marks of G.
In this way we can identify the finite G–sets with the N–linear combinations of the marks of G. As 3.5.5
describes an embedding of rings, to the product in Ω(G) there corresponds the pointwise product in Zd :
3.5.6
ε(G/Ui · G/Uj ) = µi · µj .
Restricting attention to the i-th coordinate we obtain the mapping
εi : Ω(G) → Z: X 7→ |XUi |,
and 3.5.5 says that εi is a homomorphism. Correspondingly we obtain, for any subgroup U of G, a canonical
homomorphism
εU : Ω(G) → Z: X 7→ |XU |.
But note that the number of U –invariants εU (X) = |XU | is not the desired coefficient of G/U , when we
express X as a Z–linear combination of similarity classes G/Ui of transitive G–sets:
X=
d
X
ei \\\X| · G/Ui .
|U
i=1
ei \\\X| can be obtained directly from εU (X): The equation 2.4.2 shows
But there is a certain case when |U
that the following is true:
ei is maximal such that εU (X) 6= 0 if and only if it is maximal such that
3.5.7 Lemma The subgroup U ∈ U
ei \\\X| =
|U
6 0, and in this case we have
e \\\X||NG (U )/U |.
εU (X) = |XU | = |U
Let us apply this to an example. Later on we shall use it in the proof of an astonishing theorem that leads
to important applications.
3.5.8 Example Consider the set of mappings f from G to n which are of fixed weight t (we now use as
weight the sum of the values!):
X
f (g) = t}.
nG
t := {f : G → n |
It is easy to check that f ∈ nG
t remains fixed under each g ∈ U if and only if f is constant on the right cosets
of U . Hence such an invariant f takes each of its values with a multiplicity that is divisible by |U |, therefore
G/U
|U | divides t, and the invariants f ∈ nG
t are in one-to-one correspondence with the elements of nt/|U | . The
order of this last set follows from exercise 2.3.3, and so we obtain
|G/U | + t/|U | − 1
G
εU (nt ) =
.
t/|U |
In particular, if |U | = t, then εU (nG
t ) = |G/U |, and we can apply 3.5.7, obtaining
ei , |U | = t =⇒ |U
ei \\\nG
e
U ∈U
t | = |Ui |.
In the case when the acting group is cyclic, these particular elements form a basis of the Burnside ring (see
exercise 3.5.2 for another basis):
114
Chapter 3. Weights
3.5.9 Lemma The Burnside ring Ω(Cn ) has the following Z–basis:
Cn
{nt
| t divides n}.
Proof: We denote by C(s) the unique subgroup of order s in Cn , for each divisor s of n. The matrix (nst )
of the coefficients in
X
Cn
nt =
nst Cn /C(s)
s|n
is triangular with ones along their main diagonal, and hence invertible over Z, which proves the statement.
3
We recall that
εUi (G/Uk ) = εi (G/Uk ) = mik ,
and so εU = εV if and only if U and V are conjugate. Moreover,
X
X
ek \\\X|G/Uk ) =
ek \\\X|mik ,
εi (X) = εi (
|U
|U
k
k
ek \\\X| can be obtained from the numbers εi (X), since the table of
which shows that the desired numbers |U
marks is not singular. Hence we have proved
3.5.10 Corollary We can identify the element X of the Burnside ring Ω(G) with the mapping U 7→ εU (X),
obtaining an embedding of Ω(G) into the following ring which is called the ghost ring of G:
L(G)
e
Ω(G)
:= Z∼
:= {f ∈ ZL(G) | f constant on each conjugacy class}.
e n ) of the cyclic group is equal to the Burnside ring of Cn , since Cn is abelian. Moreover,
The ghost ring Ω(C
as Cn contains just one subgroup of order t, for each divisor t of n, both rings can be identified in a canonical
way with ZT (n) , if T (n) denotes the set of divisors of n. This leads to a canonic map into the ghost ring of
G via the function card that maps U ∈ L(G) onto its cardinality:
L(G)
α: ZT (|G|) → Z∼
: f 7→ f ◦ card.
The crucial point is that this map is a ring homomorphism, which is clear from the following: First of all we
know from 3.5.8 that, if n := |G|, then, if C(|G|) denotes the cyclic group of order |G|:
C(|G|)
εU (nG
t ) = εC(|U |) (nt
).
This together with 3.5.7 implies
C(|G|)
α(nt
) = nG
t .
Finally we use that ε is a ring homomorphism, and therefore the following map must extend to a ring
homomorphism:
C(|G|)
nt
7→ nG
t .
Thus we have proved the following important result:
3.5.11 Theorem (Dress, Siebeneicher, Yoshida) If G is a finite group of order n, then
α: Ω(C(|G|)) → Ω(G): f 7→ f ◦ card,
is a ring homomorphism such that, for each finite C(|G|)–set X and any subgroup U of G, we have
εU (α(X)) = εC(|U |) (X).
3.5. The Burnside ring
115
For example,
εC(t0 ) (Cn /C(n/t)) =
n
εU (α(Cn /C(n/t))) =
n
t if C(t0 ) ⊆ C(n/t)
0 otherwise.
Hence, by the theorem,
t if t divides |G|/|U |
0 otherwise.
An application of 3.5.7 now shows that
ei \\\α(Cn /C(n/t))| =
|U
e | if t = |G/U |
|U
0
if t does not divide |G|/|U |.
3.5.12 Corollary For each divisor t of |G| there exists a G–set Xt , for which
t if t divides |G/U |
εU (Xt ) =
0 otherwise,
and therefore
ei \\\Xt | =
|U
ei |
|U
0
if t = |G/Ui |
if t does not divide |G/Ui |.
This G–set Xt is of order t.
Beautiful applications are the following proofs (due to B. Wagner and Dress/Siebeneicher/Yoshida) of Sylow’s
theorems: From 3.5.12 we derive that
X
X
ei \\\Xt ||G/Ui | ∈
t=
|U
Z · |G/Ui |.
i:t divides |G/Ui |
i:t divides |G/Ui |
This shows that t is the greatest common divisor of the indices |G/Ui | which are divisible by t. Hence, if |G|
is t times a prime power pr , there must exist a subgroup of order pr .
Dividing this equation by t and then reducing modulo p we obtain
1 ≡ |{U ≤ G | |U | = pr }| (p),
obtaining that the number of subgroups of order pr is congruent 1 modulo p. The remaining Sylow theorem
that each p–subgroup is subconjugate to each p–Sylow subgroup is left as exercise 3.5.4.
Exercises
E 3.5.1 Prove 3.5.1.
E 3.5.2 Show that also the following set is a Z–basis of Ω(Cn ):
Cn
{
| t divides n}.
t
E 3.5.3 Prove that for G–sets X of p–groups G the following congruence is true (if Ud = G):
ε1 (X) ≡ nd (X) = εG (X) (p).
E 3.5.4 Use exercise 3.5.3 in order to derive that, for any p–Sylow subgroup P and each p–subgroup U ,
there exists a subgroup V of P which is conjugate to U . (Hint: Consider a p–subgroup V , a subgroup U for
which p does not divide |G/U |, and examine εV (G/U ).)
file:f35.tex
116
Chapter 3. Weights
Chapter 4
Classes of configurations, constructive
aspects
In the preceding chapters classes of configurations were introduced, and it was shown how they can be
enumerated. The total number of configuration classes is helpful, indeed, but it is the classes and not only
their numbers which we want to have at hand, at least for the smaller cases. We therefore devote this chapter
to constructive aspects.
file:f40.tex
117
118
4.1
Chapter 4. Classes of configurations, constructive aspects
The evaluation of classes of configurations
Classes of configurations are orbits of a group G on a set Y X , and so we should introduce first of all the
standard method for the evaluation of the orbit of a particular configuration f ∈ Y X .
In the case when both |Y ||X| and |G| are very small and G or Ḡ is given as a set together with the operation
of each of its elements, then we may just apply this set to f in order to get the desired orbit G(f ). But quite
often it is so that G is given by a set of generators: G = hg1 , . . . , gr i, together with the actions of the gi on
Y X . Then, for f ∈ Y X , we can put
Ω0 := {f }, and Ωi :=
r
[
gj Ωi−1 ,
i ∈ N∗ .
j=1
It is obvious that the smallest i such that Ωi = Ωi−1 satisfies
810a
4.1.1
G(f ) = Ωi−1 .
The concrete implementation of this way of evaluating G(f ) of course may heavily depend on our knowledge
of Y X , G and G (Y X ). For example, if |Y X | = 1, then G(f ) = Ω0 , while G(f ) = X, if |G\\Y X | = 1, so that
the Cauchy-Frobenius lemma can serve as a stopping rule. The knowledge of |G\\Y X | is helpful in particular
if we are after the whole set of orbits G\\Y X , in which case we proceed with the remaining subset Y X \G(f )
correspondingly. The implementation of this method is obvious.
It is clear that a careful implementation of this procedure also yields products of the generators which lead
from f to any other element of its orbit, and which therefore form a transversal of G/Gf . Thus we can also
obtain generators of the stabilizer Gf by an application of the following fact:
8001
4.1.2 Lemma (Schreier) If U is a subgroup of G = hg1 , . . . , gr i which is finite and which decomposes as
follows into left cosets of U :
s
[
G=
hi U, where h1 = 1,
i=1
and if the mapping φ is defined by
φ: G → {h1 , . . . , hs }: g 7→ hi , if g ∈ hi U,
then U is generated by the elements φ(gi hk )−1 gi hk :
U = hφ(gi hk )−1 gi hk | i ∈ r, k ∈ si.
schreier
4.1.3 Example The symmetric group Sn can be generated by two elements, and it decomposes into two
left cosets of the alternating group:
Sn = hg1 = (12), g2 = (12 . . . n)i = An ∪ (12)An ,
and so we can apply Schreier’s Lemma to h1 = 1, h2 = (12). The corresponding mapping φ is
1
if π is even,
φ: Sn → {1, (12)}: π 7→
(12) if π is odd.
In order to obtain the generators of the alternating group according to the Schreier Lemma, we evaluate the
following matrices of elements. First of all we obtain the matrix with the gi hk as entries:
gi hk
h1 = 1
h2 = (12)
.
g1 = (12)
(12)
1
g2 = (1 . . . n) (12 . . . n) (13 . . . n)
The next step is the application of φ to its entries, where we have to distinguish if n is even or odd, obtaining
φ(gi hk )
h1 = 1 h2 = (12)
, if n is even,
g1 = (12)
(12)
1
g2 = (1 . . . n)
(12)
1
4.1. The evaluation of classes of configurations
119
and
φ(gi hk )
h1 = 1 h2 = (12)
, if n is odd.
g1 = (12)
(12)
1
1
(12)
g2 = (1 . . . n)
Inverting the elements of the last two matrices gives these matrices again, and so we need only multiply
these two matrices with the first matrix elementwise, which gives us the desired matrix containing Schreier
generators of the alternating group:
and
φ(gi hk )−1 gi hk
g1 = (12)
g2 = (1 . . . n)
h1 = 1
h2 = (12)
, if n is even,
1
1
(23 . . . n) (13 . . . n)
φ(gi hk )−1 gi hk
g1 = (12)
g2 = (1 . . . n)
h1 = 1
1
(12 . . . n)
h2 = (12)
, if n is odd.
1
(13 . . . n2)
Since we may skip the generator 1, we obtain therefrom:
h(23 . . . n), (13 . . . n)i if n is even,
An =
h(12 . . . n), (13 . . . n2)i if n is odd.
3
A direct evaluation of the stabilizer Gf will be discussed later. In the case when Y X is too big to be stored,
we can do the following in order to evaluate the set of orbits. We number the elements of Y X , being now
faced with an operation of G on n, where n := |Y ||X| . There is a canonic transversal of G\\n, consisting of
the lexicographically smallest orbit elements. So we take 1 ∈ n as the first element of the desired transversal.
Then we evaluate the minimal i ∈ n which is not contained in G(1), take this point i as the second element
of the transversal and evaluate the minimal j ∈ n that is not in G(i) and bigger than i. There are now
two cases. Either there is a g ∈ G such that gj < j, in which case j ∈ G(1), or there is no such g, so
j 6∈ G(1) ∪ G(i), and therefore j has to be taken as the third element of the transversal, and so on.
In order to evaluate this canonic transversal of G\\n in an economic way we can use a problem oriented
description of the elements of Ḡ so that not too many checks are necessary. We therefore introduce the
pointwise stabilizer of the subset k ⊆ n which we denote as follows:
CḠ (k) := {π ∈ Ḡ | ∀i ∈ k: πi = i}.
We call this group the centralizer of k in order to distinguish it from the setwise stabilizer
NḠ (k) := {π ∈ Ḡ | πk = k},
which we call the normalizer of k. (Correspondingly in the general case G (Y X ) we have CḠ (M ) and NḠ (M )
for subsets M ⊆ Y X .)1 These centralizers form the chain of subgroups
Ḡ = CḠ (0) ≥ Ḡ = CḠ (1) ≥ . . . ≥ CḠ (n − 1) ≥ CḠ (n) = {1}.
Hence there exists a smallest b ∈ n + 1 such that
4.1.4
Ḡ = CḠ (0) ≥ . . . ≥ CḠ (b) = {1}.
80
We call b the base of the action of G on n, b its length, and 4.1.4 the Sims chain of this action. Now we
consider left coset decompositions
r(i)
4.1.5
CḠ (i − 1) =
[
(i)
πj CḠ (i),
(i)
π1 = 1,
j=1
and we note that each π ∈ Ḡ can uniquely be written in terms of these coset representatives as follows:
80
120
Chapter 4. Classes of configurations, constructive aspects
(1)
(b)
π = πj1 · · · πjb .
4.1.6
80
Hence in particular the following holds:
|Ḡ| =
4.1.7
b
Y
80
r(i).
i=1
(i)
This shows that storing the πj , i ∈ b, j ∈ r(i), allows an economic way to deal with the elements of Ḡ, we
have to run through the elements of the following tree:
(1)
π1
...
. ... &
(1) (2)
(1) (2)
π1 π1
...
π1 πr(2)
. ... &
...
. ... &
8215b
4.1.8
(1)
...
...
πr(1)
...
. ... &
(1) (2)
(1) (2)
. . . πr(1) π1
...
πr(1) πr(2)
... . ... &
...
. ... &
The aim of the following pages will be to cut this tree as much as possible. A first remark concerning this
shows, that in each orbit evaluation we can cut this tree below a certain level, depending on the point the
(i)
orbit of which is to be evaluated: As πj k = k, if i > k, the following is true for the orbit of k ∈ n under G:
802
(k)
(1)
G(k) = {πj1 . . . πjk k | j1 ∈ r(1), . . . , jk ∈ r(k)}.
4.1.9
Thus the knowledge of the Sims chain considerably reduces the number of necessary checks for an orbit
evaluation. The calculation of the base is therefore very important.
ex819
4.1.10 Example Let us consider for example the action of Sp on the set of 2–element subsets of p. In order
[2]
to evaluate the base of Ḡ = Sp we first of all embed the set of 2–element subsets into n, where n := p2 , by
ordering the pairs lexicographically:
{1, 2} < {1, 3} < . . . < {1, p} < {2, 3} < . . . < {2, p} < . . . < {p − 1, p},
and by replacing the pairs correspondingly by the natural numbers 1, . . . , p2 . An easy check shows that,
according to this numbering, the following chain of canonic Young subgroups
Sp ≥ S(2,p−2) ≥ S(1,1,1,p−3) ≥ . . . ≥ S(1,...,1,2) ≥ {1}
yields via embedding the Sims chain
[2]
[2]
[2]
Ḡ = Sp[2] ≥ S(2,p−2) ≥ S(1,1,1,p−3) ≥ . . . ≥ S(1,...,1,2) ≥ {1}.
[2]
Thus p − 2 is the base for the canonic action of Sp on the set of 2–element subsets and its lexicographic
ordering. Hence in particular for p := 5 we obtain the Sims chain (with respect to the above numbering of
the pairs of points)
[2]
[2]
[2]
S5 ≥ S(2,3) ≥ S(1,1,1,2) ≥ {1}.
3
Another helpful property of the base is described in
802a
4.1.11 Lemma The elements π ∈ Ḡ are uniquely determined by their action on the base b.
Once the base b is at hand we can evaluate the orbits Ḡ(k), k ∈ b, as it is described above by 4.1.9. These
orbits can serve very well for a check if π ∈ Sn is contained in Ḡ or not: Consider π1 first. If π1 6∈ Ḡ(1),
then obviously π 6∈ Ḡ, and we can stop. If otherwise π1 ∈ Ḡ(1), then there exists a unique left coset
(1)
(1)
(1)
representative πj1 such that πj1 1 = π1 ∈ Ḡ(1), and therefore (πj1 )−1 π ∈ CḠ (1). In the next step we
(1)
check if (πj1 )−1 π2 ∈ Ḡ(2). If this is not the case, then π 6∈ Ḡ. In the other case there is exactly one orbit
(2)
(1)
(2)
representative πj2 such that (πj1 )−1 π2 = πj2 2. Hence, by induction, we obtain
4.1. The evaluation of classes of configurations
80002
121
(i)
4.1.12 Corollary Either there exist πji ∈ CḠ (i), 1 ∈ b, for which
(i−1)
(1)
(i−1)
(1)
(πji−1 )−1 · · · (πj1 )−1 πi ∈ Ḡ(i),
and
(i)
(πji−1 )−1 · · · (πj1 )−1 πi = πji i,
so that
(1)
(b)
π = πj1 · · · πjb ∈ Ḡ,
or π is not an element of Ḡ.
file:f41.tex
1 It
has been added in the first section. WSF
122
4.2
Chapter 4. Classes of configurations, constructive aspects
Transversals of configuration classes
In this section we shall restrict attention to a redundancy free construction of a transversal of the G–classes
on Y X := mn . Moreover, in order to decrease the complexity we restrict attention to G–classes of fixed
content λ = (λ1 , . . . , λm ) |= n, starting off from the canonic mapping f with this content:
fλ := (fλ (1), . . . , fλ (n)) := (1, . . . , 1, 2, . . . , 2, . . . , m, . . . , m).
| {z }
| {z } | {z }
λ1
λ2
λm
The set of all the mappings of this content will be indicated as follows:
n
mλ := {πfλ = fλ ◦ π −1 | π ∈ Sn }.
n
Since a permutation of the arguments does not change the content, this set mλ is a union of orbits of G on
mn :
n
G\\mλ ⊆ G\\mn .
n
We would like to construct a transversal of mλ . This reduces the complexity since the desired transversal
n
of G\\mn is the union of the transversals of the sets G\\mλ , taken over all the λ |= n. The crucial point is
n
now that in each orbit of G on mλ there is a unique representative which is the lexicographically smallest
element of its orbit. Therefore we may call these lexicographically smallest elements a canonic transversal
n
n
of G\\mλ . A mapping f ∈ mλ can be displayed by its λ–tabloid
i1 . . . iλ1
j1 . . . jλ2 ,
···
where we put the elements of the inverse image f −1 [{k}] of k ∈ m into the k-th row, in increasing order, so
that e.g.
f (i1 ) = . . . = f (iλ1 ) = 1, and i1 < . . . < iλ1 .
Let us consider as an example the graphs on 4 vertices which contain exactly 2 edges, so that n = 42 = 6,
m = 2, and λ = (4, 2). Before we write down all the 15 (4, 2)–tabloids in full detail, it is practical to note
that in each tabloid the first (or any other) row is uniquely determined by the remaining rows which form
the truncated tabloid. Hence in the present case we need only to display the second rows, here they are:
12, 13, 14, 15, 16, 23, 24, 25, 26, 34, 35, 36, 45, 46, 56.
According to the above arguments we have to find the orbits of S¯4 on this set. Recall that the elements
1, . . . , 6 ∈ 6 on which S6 acts, stand for the pairs {1, 2}, . . . , {3, 4} of vertices. The subgroup
[2]
S̄4 := S4 ,→ S6
is the permutation group induced by S4 = h(12), (1234)i on this set of pairs of vertices. Thus
S¯4 = h(24)(35), (1463)(25)i,
and therefore one of the two orbits of S¯4 on the set of (4, 2)–tabloids is
ω2 := {16, 25, 34},
the other orbit ω1 consists of the remaining 12 truncated tabloids. (Note that in a preprocessing calculation
we can obtain, by an application of the enumeration theory described above, that the number of graphs with
4 vertices and 2 edges is two, a result which is very helpful as stopping rule, as the present example already
shows!)
The lexicographically smallest elements of these orbits ω1 , ω2 are the tabloids corresponding to 56 ∈ ω1 and
34 ∈ ω2 . These tabloids are
1234
1256
and
.
56
34
4.2. Transversals of configuration classes
r
@
r
r
123
r
r
@
and
@
@r
r
r
Figure 4.1: The graphs with 4 vertices and 2 edges
The corresponding mappings f, f 0 ∈ 26 are
f = (1, 1, 1, 1, 2, 2), and f 0 = (1, 1, 2, 2, 1, 1).
Hence the graphs shown in figure 4.1 form a complete set of graphs on 4 vertices containing 2 edges! This
method is cumbersome, it can be used for cataloging the graphs on ≤ 7 vertices, say. We want therefore
to describe further refinements which allow to catalog the graphs with v = 10 vertices (this is in fact the
biggest v for which a complete such list is available, there are approximately 12 · 106 such graphs).
4.2.1 Example As another example we consider spin configurations on a cube for s = 1. Recall that we cu
have assumed the following labels of vertices:
4
6
1
7
5
2
3
8
Of course we have n = 8 and it is easy to map bijectively the set {−1, 0, +1} on the set 3 = {1, 2, 3}. We
know from example 3.2.4 that there 3 orbits with the content λ = (6, 1, 1) and want to find the canonic
8
transversal of this orbit. The set 3λ consists of 8!/6!1!1! = 56 mappings which can be presented as truncated
tabloids
1 1
1 2 2
8
...
...
.
2 3
7 1 3
7
Since for each oriented pair (i, j), i, j ∈ 8, i 6= j, there is an element in Oh ,→ S8 which maps it onto the pair
(j, i), then it is enough to consider action of Oh on 2-subsets {i, j} what was done in example 2.1.4. Therefore,
it is not difficult to find the the lexicographically smallest elements of these orbits and the corresponding
tabloids:
123568
123456
123567
(1̄1̄1̄01̄1̄11̄) ↔ 4
(1̄1̄1̄1̄1̄1̄01) ↔ 7
(1̄1̄1̄01̄1̄1̄1) ↔ 4
7
8
8
These configurations are illustrated in the following picture where ◦ corresponds to f (i) = −1 and • to
f (i) = 1; an empty vertex corresponds to f (i) = 0
d
d
d
d
t
d
d
d
d
d
d
d
d
d
d
t
d
d
d
t
d
Note that permuting colors on these pictures you will obtain transversals of mappings with contents (1, 6, 1)
and (1, 1, 6). However, these transversals will not be canonic. On the other hand, the canonic transversal of
configurations of the content (5, 3, 0), say, can be easily transformed into canonic transversals of mappings of
the contents (5, 0, 3) and (0, 5, 3). If it is not necessary to have canonic transversals we can always limit our
considerations to λ being a proper partition of n and next permute elements of theset m (or Y , in general).
3
Having displayed an example in full, let us go into detail in order to make the procedure of finding the
minimal representatives more efficient. Recall that we have to check if a given f is the lexicographically
smallest element in its orbit, i.e. if the following is true:
124
Chapter 4. Classes of configurations, constructive aspects
∀σ ∈ Ḡ: f ≤ f ◦ σ, for short: f ≤ Ḡ(f ).
4.2.2
M
The verification of this condition is one of the crucial parts of the whole procedure. Let us call this check
the minimality test for f . In order to carry it out in a reasonable way we recall what has been said about
Sims chains. Assume that b is the base for the action of Ḡ on n (still f ∈ mn ), so that
{1} = CḠ (b) ≤ CḠ (b − 1) ≤ . . . ≤ CḠ (0) = Ḡ,
(i)
and we assume that left coset representatives πj ar at hand such that
r(i)
[
CḠ (i − 1) =
(i)
(i)
πj CḠ (i), where π1 := 1 ∈ CḠ (i).
j=1
Thus, in order to apply the minimality test 4.2.2, we have to run through the elements of Ḡ, each of which
can uniquely be written in the form
(1)
(b)
σ = πj1 · · · πjb .
This means that in order to apply each element of Ḡ to f , say, we have to run through the following tree:
(1)
π1
...
. ... &
(1) (2)
(1) (2)
π1 π1
...
π1 πr(2)
. ... &
...
. ... &
8215a
4.2.3
(1)
...
...
πr(1)
...
. ... &
(1) (2)
(1) (2)
. . . πr(1) π1
...
πr(1) πr(2)
... . ... &
...
. ... &
It is very important to cut this tree down as much as possible. This can be done, for example, by an
application of
8216a
(i)
4.2.4 Lemma If f < f ◦ πj
CḠ (i) ≤ Ḡ:
(i)
and f (i) < f ◦ πj (i), then we have, for the orbit of f under the action of
(i)
f < CḠ (i)(f ◦ πj ).
(i)
This result allows us to cut off the branch starting with the element . . . πj
the elements of Ḡ.
from the tree 4.2.3 formed by
Besides this we can choose a suitable way in which we work through the tree 4.2.3 in order to find the smallest
element in the orbit G(f ). A remark that will lead us to a good such choice makes use of the fact that f ∈ mn
will not in general be injective. But in this case there exist τ, σ ∈ Sn \{1} such that f ◦ σ = f ◦ τ , and so we
have for each i ∈ n the equation
8217a
σ −1 CḠ (i)(f ) = τ −1 CḠ (i)(f ).
4.2.5
We can therefore subdivide Ḡ according to the Sims chain and the base b:
8218a
4.2.6
Ḡ =
b
[
(CḠ (b − i)\CḠ (b − i + 1))
[
{1}.
i=1
Correspondingly we shall run through Ḡ (in any case, whether f is injective or not) by working through the
following subsets, one row after the other, from top to bottom:
1. CḠ (b − 1)\CḠ (b),
2. CḠ (b − 2)\CḠ (b − 1),
..
.
b. CḠ (∅)\CḠ (1).
4.2. Transversals of configuration classes
125
The identity element can be left out since f ◦ 1 = f . We note that in step number b − i + 1 we are running
through the set
r(i)
[ (i)
CḠ (i − 1)\CḠ (i) =
πj CḠ (i),
j=2
(i)
if we had f ≤ CḠ (i)(f ) before, assuming that π1 = 1, for each i, so that we can always start with j = 2.
This way to procede allows to use the following remark: In the case when there exists any σ ∈ CḠ (i) for
(i)
(i)
(i)
which f ◦ πj ◦ σ = f , then πj CḠ (i)(f ) ≥ f , and so we can jump over the whole coset πj CḠ (i), by 4.2.5.
This idea can also be used in order to derive the Sims chain of the stabilizer of f . Let us indicate this
stabilizer as follows:
A := {ρ ∈ Ḡ | f ◦ ρ = f } ≤ Ḡ,
and note that, for each i ∈ b:
CA (i) ≤ CḠ (i).
4.2.7
82
(i)
A transversal of the set of left cosets CA (i − 1)/CA (i) can be obtained from the transversal {πj | j ∈ r(i)}
of CḠ (i − 1)/CḠ (i) in the following way:
4.2.8 Lemma If we denote by J(i) the set
82
(i)
J(i) := {j ∈ r(i) | ∃σj ∈ CḠ (i): πj σj ∈ A},
and if we fix, for each j ∈ J(i), such an element σj , then
(i)
{τj := πj σj | j ∈ J(i)}
is a transversal of CA (i − 1)/CA (i).
4.2.9 Corollary The set
82
{τj =
(i)
πj σ j
| j ∈ J(i)}
yields the Sims chain of the stabilizer A of f .
Another byproduct is the following test:
(i)
4.2.10 Corollary If, for each i, j, f 0 (i) 6= f ◦ πj (i), then
f 0 6∈ G(f ).
Let us now summarize what has been said so far about the minimality test which we have to carry out in
order to find the lexicographically smallest element of the orbit G(f ). We discussed the Sims chain for the
action of G on n, it leads to the tree 4.2.3, and we saw in 4.2.4 that certain cuts and jumps can be made
while running through this tree. But still this does not suffice to carry out this test in an efficient way, say
if we want to catalog the graphs on 10 vertices. We shall therefore continue by considering the evaluation of
the orbits of the centralizers CḠ (i).
file:f42.tex
82
126
4.3
Chapter 4. Classes of configurations, constructive aspects
Orbits of centralizers
We say that Ḡ ≤ Sn is compatible (with the natural order on n), if and only if the following holds:
731c
∀i ∈ n: CḠ (i)\\n consists of intervals of n.
4.3.1
So in particular Young subgroups are compatible.
732c
4.3.2 Lemma For each subgroup P ≤ Sn there exist π ∈ Sn such that πP π −1 is compatible. For short:
each subgroup P of Sn is compatible up to conjugation.
We can therefore assume without restriction that Ḡ is compatible, and hence there is also a natural way of
numbering the orbits CḠ (i) as follows:
(i)
(i)
(i)
ω1 = {1}, . . . , ωi = {i}, . . . , ωt(i) .
n
(i)
Let us consider contents of restrictions of mappings f, g ∈ mλ to these orbits. We call ωk , k < t(i), content
critical for f and g, if and only if
∀s < k: c(f ↓ ωs(i) ) = c(g ↓ ωs(i) ),
while
(i)
(i)
c(f ↓ ωk ) 6= c(g ↓ ωk ).
734c
(i)
4.3.3 Lemma If ωk is content critical for f and f 0 , then we have, for each σ ∈ CḠ (i), that
(i)
(i)
(i)
(i)
f ↓ ω1 ∪ . . . ∪ ωk 6= f 0 ◦ σ ↓ ω1 ∪ . . . ∪ ωk .
(i)
(i)
Note that ω1 , . . . , ωi are one element orbits, so f = f ◦ π and f ◦ σ = f ◦ π ◦ σ are identical there, so the
preceding lemma means first of all that in order to carry out the minimality test, if f < f ◦ σ, σ ∈ CḠ (i),
say, we can start from checking f (i) instead of starting from the very beginning f (1). Moreover the decision
(i)
(i)
if f ≤ f ◦ σ will be made at last by checking if f (m) < f (σm), where m = ω1 ∪ . . . ∪ ωk . The next step
will therefore be a discussion of content critical orbits. Let us say that f ∈ mn lies below g ∈ mn in X ⊆ n
if and only if
∀x ∈ X: f (x) ≤ g(x), for short: f ≤X g,
and we say that f lies strictly below g in X if f ≤X g and
∃x0 ∈ X: f (x0 ) < g(x0 ).
This will be abbreviated by
f <X g.
735c
(i)
4.3.4 Lemma Let π, ρ ∈ Ḡ and ωk be content critical for f := fλ ◦ π and f 0 := fλ ◦ ρ, while we put
(i)
(i)
m := ω1 ∪ . . . ∪ ωk−1 . Now, if
f ↓ m = f 0 ↓ m, and f <ω(i) f 0 ,
k
then the following is true:
f ≤ CḠ (i)(f ) ⇒ f 0 ≤ CḠ (i)(f 0 ).
Note that 4.3.4 turns out to be a generalization of 4.2.4 if we identify the point i with the orbit {i}. Finally
it should be mentioned that we can also learn from a negative result in a minimality test, which is absolutely
necessary, since otherwise we would not have the slightest chance to overcome the complexity:
4.3. Orbits of centralizers
736c
127
4.3.5 Lemma If f ◦ σ < f , so that there exists a j < n such that
∀x < j: f (σx) = f (x), while f (σj) < f (j),
we put
y := max{σx | x < j ∧ f (σx) < m}, and z := max{j, σj, y}.
n
Then, for each f 0 ∈ mλ with f 0 ↓ z = f ↓ z the following is true:
f 0 ◦ σ < f 0.
738c
4.3.6 Corollary Assume π, σ and z as in 4.3.5, and suppose that the restrictions of f and f 0 satisfy
f 0 ↓ z = f ↓ z. Then also f 0 is not a canonical representative of its orbit. The lexicographically next
candidate f 0 satisfies f 0 (z) > f (z).
file:f43.tex
128
4.4
Chapter 4. Classes of configurations, constructive aspects
Examples
Let us present all the introduced theorems in an application to concrete examples. We will consider spin
configurations on a cube with s = 1/2 and s = 1 of the contents λ = (4, 4) and λ = (2, 4, 2), respectively.
To begin with we have to determine the Sims chain for the action G 8 considering the embedding Oh ,→ S8 .
Let ρ = (234)(678) (three-fold roatation) and σ = (34)(78) (reflection) be the generators of S3 . The base
group of the wreath product S2 o S3 is generated by three reflections: τx , τy , and τz (τx ◦ τy will be denoted
τxy and so on, for short). The first (with respect to the x axis) reflection considered as an element of S8 is
displayed as (16)(25)(38)(47). Using this notation the Sims chain can be written as follows:
{1} = CG (3) < hσi = C2 = CG (2) < hσ, ρi = S3 = CG (1)
< hσ, ρ, τx i = Oh = CG (0).
Therefore, the set {1, 2, 3} is the base of the considered action and each element of Oh can be written as a
product τα ρβ σ γ (α = 0, x, y, z, yz, xz, xy, xyz with τ0 = 1 and β = 0, 1, 2, γ = 0, 1). Below one branch of
the corresponding tree is presented:
.
τxy
.&
τxy τxy σ
τxy
↓
τxy ρ
.&
τxy ρ τxy ρσ
&
τxy ρ2
.
.&
τxy ρ2 τxy ρ2 σ
Each configuration of the content (4, 4) can be given as a truncated tabloid j1 j2 j3 j4 , as an 8–tuple (f (1) . . . f (8)),
where we will replace ∓1/2 by ∓ for the sake of simplicity (we assume that “–”<“+”) or graphically as a
cube with
circles and bullets under the assumption that “◦”and “•”correspond to ∓1/2, respectively. There
are 84 = 70 such configurations which, according to 3.1.11, form 6 orbits. The first, canonic, mapping is
given as
f(4,4) = (− − − − + + ++) ↔ 5678.
8
After considering next 10 configurations (in the lexicographic order in {−, +}(4,4) ) we can find the following
four elements of the canonic transversal T(4,4) :
(− − − + − + ++) ↔ 4678,
(− − − + + + +−) ↔ 4567,
(− − + + − − ++) ↔ 3478,
(− − + + − + −+) ↔ 3468.
Therefore, we have to determine the sixth, i.e. the last, one. Recall that we will be working in three steps:
• CG (2) \ CG (3), with the coset representative
(3)
π2 = σ = (34)(78);
• CG (1) \ CG (2) with the representatives
(2)
π2 = ρ = (234)(678)
(2)
π3 = ρ2 = (243)(687);
• CG (0) \ CG (1) with the representatives
(1)
π3 = τy = (17)(28)(35)(46)
π4 = τz = (18)(27)(36)(45)
(1)
π5 = τyz = (12)(34)(56)(78)
(1)
π7 = τxy = (14)(23)(58)(67)
π2 = τx = (16)(25)(38)(47)
π6 = τxz = (13)(24)(57)(68)
(1)
π8 = τxyz = (15)(26)(37)(48)
(1)
(1)
(1)
4.4. Examples
129
The next candidate is (− − + + − + +−), which fails the minimality at the first step since (− − + + − + −+)
is smaller (it is exactly the previous configuration). So we go further and consider f = (− − + + + − −+).
If this configuration will not be minimal, then we can also skip the next one, (− − + + + − +−), because it
certainly fails the minimality test (due to the action of σ = (34)(78)). Our new candidate fulfils f < f ◦ σ,
then we can start from CG (1) \ CG (2). We can cut off branches ρ i ρ2 since
f ◦ ρ2 = (− + − + + + −−) > f.
f ◦ ρ = (− + + − + − +−) > f,
Now we have to take into account reflections τα . The branches τz , τxz , τxy , and τxyz can be cut off. The
other 3 configurations, f ◦ τx , f ◦ τy and f ◦ τyz , are as follows:
(− + + − − − ++),
(− + + − + + −−),
(− − + − − + +−).
The first two do not fulfil the condition f ◦ τ (1) > g(1), but we have f ◦ τ > g. The third one does not fulfil
even this condition, what indicates that f is not minimal, since f > f ◦ τyz . Therefore, f has failed our test
and we have to check the next candidate (skipping one, as it was mentioned above): f = (− − + + + + −−).
Considering this element we will also determine the Sims chain for its stabilizer Gf = A.
Since f ◦ σ = f then the transversal CA (2)/CA (3) = {1, σ}. The rotations ρ and ρ2 yield
f ◦ ρ = (− + + − + − −+), and f ◦ ρ2 = (− + − + + − +−),
so these branches can be cut off and, moreover, we obtain that J(2) = {1}, because there is no such σ ∈ CG (2)
that ρσ or ρ2 σ stabilizes f . For reflections τα we obtain
f ◦ τ0 = f ◦ τy = f ◦ τz = f ◦ τyz
f ◦ τx = f ◦ τxz = f ◦ τxy = f ◦ τxyz
= (− − + + + + −−),
= (+ + − − − − ++).
Elements in the first row form the transversal of CA (0)/CA (1), while those in the second one show that the
corresponding branches can be cut off. Therefore we found the sixth (represented by the truncated tabloid
3456) element of the transversal T(4,4) and determined its stabilizer A. The Sims chain of A is given as
{1} = CA (3) < hσi = CA (2) = CA (1) < hσ, τy i = CA (0) ' D4
(note that D4 ' S2 o S2 ) and the corresponding tree can be drawn as follows
1
τy
↓
↓
1
τy
.&
.&
1 σ τ y τy σ
τz
↓
τz
.&
τ z τz σ
τyz
↓
τyz
.
.&
τyz τyz σ
8
It is not difficult to check that the element, which was considered as the last one, is the 15th in {−, +}(4,4)
with respect to the lexicographic order. Therefore, in order to find 6 representatives out of 70 elements we
had to check only about 20 % of them (recall that the first element, fλ , does not need checking due its
defintion). The obtained representatives are presented below:
d
t
t
t
t
t
t
d
t
t
t
t
t
d
t
d
t
d t
d d
d d
d
d
t
t
d
d
d
t
d
d
t
d
d
d
d
t
t
d
d
t
t
d
t
d
t
Let us consider briefly the case s = 1. We know from example 3.2.4 that there are 45 different contents
and 267 orbits. However, we are going to consider only orbits with magnetization M = 0, so we have to
determine 43 representatives. These orbits can be divided into five subsets with different contents and the
formulas in example 3.2.4 yield that there are
1
3
16
17
6
orbit with the content
orbits with the content
orbits with the content
orbits with the content
orbits with the content
080
161
242
323
404
130
Chapter 4. Classes of configurations, constructive aspects
The first case is trivial and we have T(0,8,0) = {(00000000)}. The last case is identical with this solved
above (4 spins up and 4 down). In the second case it is easy (cf. example 4.2.1) to find that T(1,6,1) =
{(1̄0000001), (1̄0001000), (1̄0010000)} (1̄ stands for –1). Therefore, we have to find 16 representatives out of
8!
2!4!2! = 420 configurations in the third case and 17 out of 560 in the fourth one. Sometimes recursive methods
are very helpful, but the recursion with respect to M needs determination of transversals TM =−8 , . . . , TM =−1 ,
and TM =0 , so, if we are interested only in TM =0 , it is a long way (and will be longer for larger spins s). It
is also possible to consider recursive methods with respect to s, but, as we shall see in the next sections,
these methods do not keep fixed M = 0, so we would have to consider all transversals TM =−4 , . . . , TM =0
for s = 1/2, hence, once more, it is not the most convenient method. Therefore, we are going to find the
transversal TM =0 (for s = 1) directly (moreover, we have just determined 10 elements of this transversal).
We show only a few steps in determination of the canonic transversal T(242) in order to illustrate applications
of lemma 4.3.5 and corollary 4.3.6. Strating from the canonic configuration
f(242) = (1̄1̄000011)
after some steps we have to consider f = (1̄1̄100001). Note that f ◦ σ = (1̄1̄010010) < f then f has to be
rejected. However applying 4.3.6 we can learn which configurations can be skipped. The configuration f ◦ σ
is less than f at j = 3. Since σ1 = 1, σ2 = 2 and f (σ1) = f (σ2) = 1̄ < 1 then y = 2. On the other hand
σj = σ3 = 4, hence the maximum of j, σj and y is z = 4. Therefore, to obtain the next candidate we have
to change f (4) and to consider f 0 = (1̄1̄110000), which occurs to be the element of T(2,4,2) . The next in the
8
lexicographic order in {1̄, 0, 1}(2,4,2) is f = (1̄01̄00011). It is easy to note that f ◦ ρ = (1̄1̄000110) < f and
z = σ2 = 3. We again can skip some configurations and consider the next one to be f = (1̄001̄0011), for
which we obtain f ◦ σ < σ, so we have to change f (4). As the result we have f 0 = (1̄0001̄011), which is the
minimal element in its orbit.
In the similar way we can obtain the canonic transversal T(3,2,3) . Since the transversals T(0,8,0) , T(1,6,1) , and
T(4,0,4) are easy to construct we present below the other two only.
4.4. Examples
131
d
t
d
t
d
d
t
d
t
d
d
t
d
t
t
d
t
d
t
d
d
t
t
t
d
d
t
t
d
t
d
t
d
t
t
d
file:f44.tex
t
t
d
d
t
t
d
t
d
d
t
d
t
t
d
d
t
d
d
t
t
d
t
d
t
t
t
d
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t
d
d
t
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d
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d
t
d
t
d
t
t
d
t
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t
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t
d
d
d
d
d
t
t
d
d
t
d
d
d
t
t
d
d
t
d
d
d
t
d
t
t
t
t
d
d
d
d
d
d
d
d
d
t
t
t
t
d
d
t
d
t
d
t
d
t
t
d
d
d
t
t
d
t
d
d
t
t
d
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d
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d
d
d
d
t
t
d
d
t
t
t
t
t
d
t
132
4.5
Chapter 4. Classes of configurations, constructive aspects
Generalization
In the previous sections we presented methods which have occured very useful in determination of configuration transversals. However, the investigated examples showed that there are still some problems, which
have to be solved in order to obtain more efficient algorithms.
Considerations presented below are not, in fact, related to genaralization, but I have no idea where they can be
included. There are at least 4 possibilities:
1. Leave it as it is;
2. Remove all, since it is discussion on recursive methods for Young diagrams (tableaux?) and — I think — they
are known;
3. Move this material to the end of the previous section (Examples);
4. Make a separate section for this recurence.
There is another problem (not so important). The lexicographic order in [−s, +s] is −s < −s + 1 < . . . < s,
so this set is order isomorphic with the set 2s + 1 = {0, 1, 2, . . . , 2s}. Assuming this isomorphism the smallest
element in [−s, +s]n is (−s, −s, . . . , −s) and other configurations are obtained by raising spin projections.
This leads, however, to the representation by Young tableaux, where the left-most columns correspond to λ2s
(λ = (λ0 , λ1 , . . . , λ2s ) is a content) and, accordingly, the right-most columns correspond to λ0 , so the order is
reversed in comaprision with the natural one. Moreover, the recursive procedure, which is described below, leads
to cutting off the right-most columns. It means that λ = (λ0 , λ1 , . . . , λ2s ) 7→ (λ1 , . . . , λ2s ) = (λ00 , . . . , λ02s0 )
with s0 = s − 1/2. This can be changed, i.e. the left-most columns correspond to λ0 and we cut off columns
corresponding to λ2s so λ = (λ0 , λ1 , . . . , λ2s ) 7→ (λ0 , . . . , λ2s−1 ). However, the method will be self-consistent if
we assume the order of projections given as s < s − 1 < . . . < −s. Thes means, among others, that the canonic
transversals in the previous sections consist of the maximal element in each orbit. I would like to avoid such order
in [−s, +s] and to keep methods consistent. Therefore, I have to cut off λ0 columns etc.
Considering spin configuration with given magnetization we have noticed that for a given M there are some
corresponding contents. In the case of small spin numbers, s = 1/2 or 1, it is easy to determine all contents
λ0 = (λ0−s , λ0−s+1 , . . . , λ0s ) fulfilling the condtions (sum rules)
X
jλ0j = M
j∈[−s,+s]
X
and
λ0j = n.
j∈[−s,+s]
For large spin number we can find necessary contents considering generating functions for the genearl weight
and for the weight determined by the magnetization. However, if we are interested only in a few possible
values of M , −ns ≤ M ≤ ns, then we have to consider very long polynomials in order to use only some of
its sumands. Applying a bijection [−s, +s] → m + 1 = {0, 1, . . . , m}: j 7→ j + s for m = 2s and relabelling
accordingly contents (i.e. λ = (λ0 , . . . , λm ) and λi = λ0i−s ) the above sum rule can be rewritten as follows
X
X
X
(i − s)λi =
iλi − ns =
iλi − ns = k − ns
i∈m+1
i∈m+1
i∈m
and k = M + ns. It is evident then this sum rule corresponds (up to the constant term ns) to partitions of
k into no more than m = 2s parts no longer than n. We consider such partitions in example 1.5.8 and we
obtain formula for their number (cf. 1.5.10) since there exists a natural bijection between these partitions
and orbits of Sm o Sn acting on 0-1-matrices m × n. Therefore, there also exists bijection between these orbits
and all possible contents λ (and so λ0 , too) for n spins s = m/2. It is evident that the canonic configuration
fλ0 = (−s, . . . , −s, −s + 1, . . . , −s + 1, . . . , s, . . . , s)
| {z } |
{z
}
| {z }
λ0−s
represents an orbit of the symmetric group Sn
following matrix
λm
z }| {

1
1...1

2
1...1
 .
..
 ..
.

m − 11...1
m
1...1
λ0−s+1
λ0s
acting on [−s, +s]n (cf. also 3.4.19) and it corresponds to the
λm−1
...
z }| {
λ1
λ0
z }| { z }| {
1...1
1...1
..
.
...
...
.
..
1...1
0...0
...
...

1...1 0...0

0...0 0...0
..
.. 
.
.
. 


0...0 0...0
0...0 0...0
4.5. Generalization
133
We have to reverse the order of λ’s or count 0’s instead of 1’s. According to the substitution m = 2s we
have that there are 22ns matrices and (2s + 1)n spin configurations so for s = 0, 1/2 these numbers are
equal, but for larger spin numbers there are more matrices than configurations (since 22s > 2s + 1). We can
find surjection 2m×n → [−s, +s]n but it is a bit atrificial and really not necessary in our considerations.2
A problem, which we want to solve, consists in determination of all proper partitions with given n, s (i.e.
m = 2s) and M (i.e. k = M + ns). It is easy in some special cases as one can see in the following examples
4.5.1 Examples
ex
• If M = ns then k = 2ns = mn and all entries of the corresponding matrix have to be equal to 1. On
the other hand, if M = −ns then k = 0, so all entries are equal to 0.
• It is also very easy to solve this problem for M = −ns + 1 and M = ns − 1, i.e. for k = 1 and mn − 1,
respectively. Note that the time-reversal symmetry corresponds to the transposition 0 ↔ 1 followed
by neccesary permutations of columns and of entries in each column.
• For s = 1/2 (m = 1) we have to consider one-row matrices with k = 0, 1, . . . , n ones and, of course,
there is only one such matrix for a given k (or a given magnetization M = k − n/2). It confirms the
fact that for s = 1/2 there is only one content corresponding to given magnetization.
• When n = 2 then (for a given 0 ≤ k ≤ 4s) we have to find all partitions α ` k into no more than 2s
parts no longer than 2 which. For example, s = 2 (m = 4) and M = −1 (k = 3) yield the following
matrices (and, therefore, representatives of S2 \\M =−1 [−2, 2]2 ; the canonic configurations are written
in the parentheses and the corresponding contents c(f, j) with j ∈ [−2, 2] are given in the square
brackets):




11
10
(−1, 0)
(−2, 1)
1 0
1 0
 ↔ [0, 1, 1, 0, 0] .
 ↔ [1, 0, 0, 1, 0] , 

00
10
00
00
3
It is not difficult to notice that the set of all proper partitions for given n, m, and k can be decomosed into
subsets with fixed lenght of the first row, i.e. α1 . Of course, we have max(0, k/m) ≤ α1 ≤ min(n, k). All
matrices in a subset with given α1 have the following form:


1 1 ...1 0...0
 1 1 ...
0...0 



.
..
..


.
.
1 1 ...
0...0
The lower left part of such matrices contains k 0 = k − α1 1’s distrubuted in m0 = m − 1 rows of the lenght
n0 = α1 and, therefore, it corresponds to a proper partition of k 0 into no more than m0 parts no longer
than n0 . Since in many physical applications the number of crystal nodes, n, is greater than the number of
one-node states, m + 1, then this recursive method has the advantge that n decrease faster than m leading
in a few steps to partitions which can be easily determined (e.g. those of example 4.5.1). Note that this
procedure determines only contents, so the symmetry group of a lattice (consisting of n nodes) is not taken
into account. It is obvious since, in fact, we consider the natural action of Sn on n.
4.5.2 Example Let us find all contents corresponding to M = 0 when s = 3/2 and n = 8 (for example we s3
are again interested in spin configurations on a cube). It means that we have to find all proper partitions of
k = 0 + 8 · 23 = 12 into no more than 2 · 32 = 3 parts no longer than 8. The possible values of α1 are: 4, 5, 6,
7, and 8, so in the second steps we have to consider the following partitions into no more than 2 parts:
• of k 0 = 8 no longer than 4;
2I
think that the last two sentences can be omitted.
134
Chapter 4. Classes of configurations, constructive aspects
Table 4.1: Contents corresponding to M = 0 for 8 spins
a ∈ 23×8


1111 0000 



 1111 0000 
 1111 0000 
1111 1 000 


 1111 0 000 

1110 0 000
11111 000 


 11111 000 

 11000 000 
111 111 00 


 111 000 00 

 111 000 00 
1111 11 00 


 1111 00 00 

 1100 00 00 
11111 1 00 


 11111 0 00 

10000 0 00
111111 00 


 111111 00 

 000000 00 
111 1111 0 


 111 0000 0 

 110 0000 0 
1111 111 0 


 1111 000 0 

 1000 000 0 
11111 11 0 


 11111 00 0 

 00000 00 0 
11 111111 


 11 000000 

 11 000000 
111 11111 


 111 00000 

 100 00000 
1111 1111 


 1111 0000 

0000 0000
total
λ
fλ
3
2
|Oh \\λ [ 23̄ , 32 ]8 |
[4, 0, 0, 4] ( 3̄2 , 3̄2 , 3̄2 , 3̄2 , 23 , 32 , 32 , 32 )
6
[3, 1, 1, 3] ( 3̄2 , 3̄2 , 3̄2 , 1̄2 , 21 , 32 , 32 , 32 )
30
[3, 0, 3, 2] ( 3̄2 , 3̄2 , 3̄2 , 12 , 21 , 12 , 32 , 32 )
17
[2, 3, 0, 3] ( 3̄2 , 3̄2 , 1̄2 , 1̄2 , 21̄ , 32 , 32 , 32 )
17
[2, 2, 2, 2] ( 3̄2 , 3̄2 , 1̄2 , 1̄2 , 21 , 12 , 32 , 32 )
68
[2, 1, 4, 1] ( 3̄2 , 3̄2 , 1̄2 , 12 , 21 , 12 , 12 , 32 )
22
[2, 0, 6, 0] ( 3̄2 , 3̄2 , 12 , 12 , 21 , 12 , 12 , 12 )
3
[1, 4, 1, 2] ( 3̄2 , 1̄2 , 1̄2 , 1̄2 , 21̄ , 12 , 32 , 32 )
22
[1, 3, 3, 1] ( 3̄2 , 1̄2 , 1̄2 , 1̄2 , 21 , 12 , 12 , 32 )
30
[1, 2, 5, 0] ( 3̄2 , 1̄2 , 1̄2 , 12 , 21 , 12 , 12 , 12 )
6
[0, 6, 0, 2] ( 1̄2 , 1̄2 , 1̄2 , 1̄2 , 21̄ , 1̄2 , 32 , 32 )
3
[0, 5, 2, 1] ( 1̄2 , 1̄2 , 1̄2 , 1̄2 , 21̄ , 12 , 12 , 32 )
6
[0, 4, 4, 0] ( 1̄2 , 1̄2 , 1̄2 , 1̄2 , 21 , 12 , 12 , 12 )
6
236
• of k 0 = 7 no longer than 5;
• of k 0 = 6 no longer than 6;
• of k 0 = 5 no longer than 7;
• of k 0 = 4 no longer than 8.
We can proceed further according with the described procedure but it is easy find all necessary partitions “by
hand”. We obtain 13 different partitions, so there are 13 contents corresponding to M = 0. Example 3.2.4
provides us with the number of orbits of Oh in this case: there are 236 orbits and considering the general
weight we can find how many of them correspond to each content. All results are given in table 4.1 where
we gathered partitions represented by 0-1-matrices 3 × 8 (vertical lines denote recursion steps), contents,
canonic configurations and numbers of orbits with a given content.
3
Let us return to the Sims chain now. The presented examples were not very complicated but they showed
that efficiency of our method strongly depends on labelling of elements in X. For example, we should avoid
trivial decompositions in the Sims chain, i.e. CḠ (i) should be a proper subgroup of CḠ (i − 1). On the other
hand, this subgroup ought to be as large as possible to decrease the number of representatives at each level.
Therefore, it is very important to investigate the action G X and the lattice of subgroups L(G) before the
4.5. Generalization
135
action G Y X . In the case when X is a set of mappings, for example a finite crystla lattice Λ = ZnN , we can use
Burnside’s Lemma to classify x ∈ X by their stabilizers. However, according to corollary 2.4.11 a subgroup
U ≤ G has to fulfil a strong condition to be a stabilizer. For example, the octahedral group Oh ' S2 o S3
consisting of 48 elements has 98 subgroups (including the improper ones) group in 33 conjugacy classes,
but only the group S30 (the complement of the base group S2∗ ) and its subgroups can appear as centralizers,
what we saw in the discussed example. The base group S2∗ = S2 × S2 × S2 acts in the regular way on
cube vertices, so each U < Oh such that U ∩ S2∗ 6= {1} can not be a centralizer. If we take into account
that |Oh | = 48 = 24 · 3 then it is easy
that there may be a chain of subgroups U ≤ Oh consisting
Qr to see
i
,
where pi are different primes, then there may be a chain of
of 6 subgroups (in general, ifP|G| = i=1 pα
i
r
subgroups consisting of 1 + i=1 αi elements including improper ones). Note, that this problem is very
important in applications to solid state investigations, because we deal with G being a space group of Λ
and this group contains the translation group T , which acts in the regular way on Λ. Therefore, increasing
the order of Λ we will not be able to consider longer Sims chains (there is one exception since for N > 2
subgroups generated by reflections τα can be centarlizers, cf. example 1.6.13) and at the last level we will
have to consider N n representatives of cosets determined by a point group P in a space group S = T o P .
4.5.3 Example In order to solve this problem we start with detailed consideration of a simple (but not si
too simple) example. Let X = 6 = {0, 1, 2, 3, 4, 5} be a regular orbit of Z6 ' C6 (the action of Z6 on 6 is
determined by addition modulo 6). The first things we have to do is to forget about the “natural” order
0 < 1 < . . . < 5 and to consider lattice of subgroups L(Z6 ). It is clear that we have two possible chains of
subgroups: Z6 > Z3 > Z1 and Z6 > Z2 > Z1 . Choosing one of them we determine a new order in X = 6
corresponding to the decomposition of Z6 into left cosets. For example, the first chains leads to the following
order 0 < 2 < 4 < 1 < 3 < 5 and we have the following tree of coset representatives:
.
0+0
0
↓
0+2
&
0+4
.
1+0
1
↓
1+2
&
1+4
Let us consider mappings f : 6 → {−, +}, now.
We want to determine the canonic transversal corresponding to the content λ = (4, 2). The canonic mapping
(− − − − ++) belongs to Tλ so we start from the next, i.e. f = (− − − + −+). Elements 2 ∈ Z6 leads to the
mapping
(− − − − ++),
so chosen f fails the minimality test. It is obvious that the next mapping (− − − + +−) is not minimal in
its orbit, too. Now we have to test (− − + − −+), for which we have, under action of 2 and 4,
(− + − − +−)
and
(+ − − + −−).
We see that this mapping passed the first minimality tests so now let us consider action of 1 ∈ Z6 , which
leads to
(− − + − +−).
file:f45.tex
136
4.6
Chapter 4. Classes of configurations, constructive aspects
Recursive construction
We have seen how symmetry classes of spin–configurations can be defined as orbits of cyclic groups Cn on
sets of spin–configurations
[−s, +s]n .
Moreover, we saw how they can be enumerated. It therefore remains to discuss the most serious question,
how they can be constructed, which means to evaluate a transversal, a complete system of representatives of
these sets of symmetry classes.
Besides direct methods of evaluating a transversal of the symmetry classes with prescribed content, there
exists also a recursive method, where the recursion is on the cardinality of the range. We are going to
describe this next. It is based on the following observation:
4.6.1 Lemma Assume an action G X, and that B ⊆ X is a block or, as it is sometimes called a system
of imprimitivity (which means that B is mapped under g ∈ G either onto B or onto a subset gB which is
disjoint with B). Then the following is true:
• For each orbit ω ∈ G\\X we have
ω ∩ B 6= ∅ ⇐⇒ ω ∩ gB 6= ∅,
which means that B and gB intersect with the same orbits.
• For each b ∈ B, g ∈ G, gb ∈ B is true if and only if
g ∈ NG (B) := {g ∈ G | gB = B}.
Therefore, if
X=
[
Bi ,
i∈I
a disjoint union of blocks, and J ⊆ I such that
T (G\\{Bi | i ∈ I}) = {Bj | j ∈ J}
is a transversal of G on the set of these blocks, then a transversal T (G\\X) is the following union of transversals:
[
T (G\\X) =
T (NG (Bj )\\Bj ),
j∈J
where T (NG (Bj )\\Bj ) is an arbitrary transversal of NG (Bj )\\Bj .
This result shows that the direct evaluation of a transversal can be replaced by successive evaluations of
blocks, their normalizers and transversals of the orbits of the normalizers on the blocks. Hence we can reduce
the complexity very much by using blocks. An interesting method systematically to obtain blocks can be
derived from
4.6.2 The Homomorphism Principle Assume that G acts on the sets Xi , i = 1, 2, and that ϕ: X2 → X1
commutes with the action:
ϕ(gx2 ) = gϕ(x2 ), for each g ∈ G, x2 ∈ X2 ,
then each inverse image ϕ−1 (x1 ), x1 ∈ X1 , is a block. Moreover, the normalizer of this block is the stabilizer
of x1 :
NG (ϕ−1 (x1 )) = Gx1 .
This is easy to check but very important to remark since it is essential for
4.6. Recursive construction
137
4.6.3 Surjective Resolution Assume that G acts on the sets Xi , i ∈ m, and that we are given surjective
mappings
ϕi+1 : Xi+1 → Xi , for each i ∈ m − 1,
that commute with the action:
ϕi+1 (gxi+1 ) = gϕi+1 (xi+1 ), for each i ∈ m − 1, g ∈ G, xi+1 ∈ Xi+1 .
Then we can obtain a transversal T of G\\Xm by successively working backwards, starting by evaluating a
transversal of G\\X1 and the stabilizers of its elements, and then evaluating the inverse images ϕ2 (x) of the
elements x in this transversal, as well as the transversals Gx \\ϕ−1
2 (x), and so on, in accordance with 4.6.1.
We would like now to apply this to symmetry classes in order to obtain a recursive evaluation of a transversal,
using recursion on |Y |.
4.6.4 Example Let us consider the Heisenberg magnet with n nodes and spin s. Hence
Y X = [−s, +s]n ,
where
X := n := {1, 2, . . . , n}
and
Y := [−s, +s] := {−s, −s + 1, . . . , s − 1, s},
is the set of spin configurations. The symmetry group of X is G and it acts on Y X = [−s, +s]n in the
canonic way.
The Surjective Resolution method 4.6.3 can be applied since the following mapping ϕs commutes with the
action of G on the sets [−s, +s]n :
ϕs : [−s, +s]n −→ [−s + 1/2, s − 1/2]n ≡ [−s0 + s0 ]n : f 7→ f 0 ,
where s0 = s − 1/2 (so 2s0 + 1 = 2s) and f 0 is defined by
f (i) + 1/2 for f (i) ∈ [−s, s − 1]
f 0 (i) :=
s − 1/2
otherwise.
We can therefore start from X0 := {0}n , which consists of the single and constant mapping i 7→ 0, i ∈ n.
The stabilizer is G, and the inverse image is the whole set X1/2 := {−1/2, +1/2}n . We therefore have to
evaluate in this second step a transversal T1/2 of G\\[−1/2, +1/2]n . Assume that this is done, and let f1/2
denote on element of T1/2 , Gf1/2 its stabilizer. It remains to evaluate in the third step, for each such f1/2
and its stabilizer a transversal T1 of
Gf1/2 \\ϕ−1
1 (f1/2 ).
This inductive step can be carried out as follows. Two elements f1 and f10 of ϕ−1
1 (f1/2 ) can differ only on
−1
the inverse image f1/2
(1/2) of the point 1/2, and the values there can only be 0 or 1. We therefore need
only to evaluate a transversal T of
f −1 (1/2)
Gf1/2 \\{0, 1} 1/2
The desired transversal T1 then consists of the mappings f100 ∈ [−1, 1]n such that
f100 |f −1 (1/2) ∈ T, and f100 |n\f −1 (1/2) = f1/2 |n\f −1 (1/2) .
1/2
1/2
1/2
Let n = 4, s = 1, and G = D4 . Then we have
X0 = {(0000)}
and the transversal T is given by
T1/2 = {( 21̄ , 1̄2 , 1̄2 , 1̄2 ), ( 12 , 1̄2 , 1̄2 , 21̄ ), ( 12 , 12 , 1̄2 , 1̄2 ),
E
138
Chapter 4. Classes of configurations, constructive aspects
( 21 , 1̄2 , 12 , 21̄ ), ( 12 , 21 , 12 , 1̄2 ), ( 12 , 21 , 21 , 12 )}.
The stabilizers, in the same order, are as follows (cf. 2.4.9)
D4 , S2 , S20 , D2 , S2 , D4 ,
The corresponding inverse images are given bellow:
{(1̄1̄1̄1̄)};
{(111̄1̄), (101̄1̄), (011̄1̄)∗ , (001̄1̄)};
{(11̄1̄1̄), (01̄1̄1̄)};
{(11̄11̄), (11̄01̄), (01̄11̄)∗ , (01̄01̄)};
{(1111̄), (1101̄), (1011̄), (1001̄), (0111̄)∗ , (0101̄), (0011̄)∗ , (0001̄)};
{(1111), (1110), (1101)∗ , (1100), (1011)∗ , (1010),
(1001)∗ , (1000), (0011)∗ , (0010)∗ , (0001)∗ , (0000)}.
All the transversals T are gathered in table 4.2. The transversal T1 consists of the configurations f1 ∈ [−1, 1]n ,
which are not labelled by an asterisk ∗ in the above formula. So we obtain 21 representatives what agrees
with 2.4.9.
Table 4.2: Transversals T for n = 4 and s = 1
set of orbits
D4 \\∅
S2 \\{0, 1}1
S20 \\{0, 1}2
D2 \\{0, 1}{1,3}
S2 \\{0, 1}3
D4 \\{0, 1}4
transversal
∅
{(0), (1)}
{(00), (10), (11)}
{(00), (10), (11)}
{(000), (010), (100), (101), (110), (111)}
{(0000), (0001), (0011), (0101), (0111), (1111)}
3
file:f46.tex
4.7. Orderly generation
4.7
139
Orderly generation
The
method
of
Surjective
Resolution
can
be
combined
with
the
method of orderly generation as described by R. C. Read. This way of generation is based on the fact
that total orders on X and Y induce a canonic total order on Y X , so that a canonic transversal
T> (G\\Y X ),
consisting of the biggest elements of the orbits does exist. Moreover, we can break this problem into pieces,
if weights are at hand.
4.7.1 Example Recall that spin configurations on n nodes and spin number s form the set
[−s, +s]n
and that an action of G ≤ Sn can be investigated (if we consider a linear chain then G = Dn ). As the sets
n and [s, −s] are totally ordered, so is the set [−s, +s]n . This is in fact the lexicographic order ≤ on the set
of sequences f (1)f (2). . .f (n), f (j) ∈ [−s, +s]. It means that we shall write
f < f0
if, for a suitable i < n, f (j) = f 0 (j) for j ≤ i and f (i + 1) < f 0 (i + 1) (the values of f on the j > i + 1
are irrelevant). For example, when G = D4 , n = 4 and s = 1, the configuration (1, 0, 1̄, 1̄) is the canonic
representative of its orbit, but (1̄, 0, 1, 1̄), belonging to the some orbit, is not the canonic representative.
The problem of orderly generation now means the following. We would like to find an economic way of
generating configurations f ∈ [−s, +s]n in such a way that not too many sequences are generated which do
not correspond to the desired set of canonic representatives. Moreover, we should like to do this for each
value of
X
M (f ) :=
f (i)
i∈n
separately, by increasing this number by 1, which means by successively increasing the total magnetization
of configurations. Of course, all the possible values of M form the set [−ns, +ns].
We start from the configuration (−s, −s, . . . , −s), which forms the single and therefore also canonic transversal of the configurations with m = −ns:
T>−ns = {(−s, −s, . . . , −s)}.
Now we will evaluate T>−ns+1 , then T>−ns+2 , and so on, in the following way. Assume that T>M is at hand.
Then, in order to obtain T>M +1 , we shall apply to each member of T>M the following procedure:
4.7.2 The augmentation: We scan the entries of f ∈ T>M from the right to the left, and we increase
exactly one entry, according to the following rules:
• We look for the rightmost entry which is bigger than the smallest possible entry (which is −s in the
spin configurations’ case), we increase this element by one, if it is smaller than the maximal possible
entry, otherwise we do not change this vector but we go to the next step.
• The remaining configurations are obtained by increasing by 1 in turn each of the following entries,
which are equal to the smallest possible entry, proceeding from left to right.
Having done this for each element f ∈ T>M , we have obtained a set of elements which contains the desired
canonic transversal T>M +1 . It has to be checked carefully, the canonic configurations have to be kept, the
others have to be thrown away before going to the next step.
Here is a concrete example: We would like to evaluate the canonic transversal of [−s, s]n , for n = 4, s = 1
and G = D4 . The augmentation procedure runs as follows: We start from
T>−4 = {(1̄1̄1̄1̄)}
and the augmentation process runs as follows:
140
Chapter 4. Classes of configurations, constructive aspects
1. At first we note that (1̄1̄1̄1̄) does not contain any entry bigger than −1, and so it is only the second
item of the augmentation that applies. It yields the four vectors with exactly on zero entry, all other
entries are −1. The only canonic representative is, of course, (01̄1̄1̄), and so
T>−3 = {(01̄1̄1̄)}.
2. The second step gives the configurations (11̄1̄1̄), according to the first step in the augmentation, and,
according to the second step, the configurations (001̄1̄), (01̄01̄), and (01̄1̄0). The last configuration is
not canonic, and so
T>−2 = {(11̄1̄1̄), (001̄1̄), (01̄01̄)}.
3. In the third step we get the canonic representatives (101̄1̄) and (11̄01̄) from (11̄1̄1̄). The second and
the third element of T>−2 do contribute only one canonic configuration, namely (0001̄). Hence
T>−1 = {(101̄1̄), (11̄01̄), (0001̄)}.
4. The next step gives three canonic representatives from the first element of T>−1 , one from the second
one, and one new — (0000) — from the last, and hence
T>0 = {(111̄1̄), (1001̄), (101̄0), (11̄11̄), (0000)}.
5. In the fifth step we get
T>1 = {(1101̄), (1011̄), (1000)}.
6. The two next steps gives
T>2 = {(1111̄), (1100), (1010)}
and
T>3 = {(1110)}
from which we obtain in
7. the final step that
T>4 = {(1111)}.
Note that we obtained transversal different from that one in example 4.6.4 — in T>0 we have (101̄0), whereas,
in the previous case, this orbit was represented by (0101̄). It is due to the fact that above we have been
looking for the canonic transversal.
3
In the above process we have generated 21 configurations from the canonic transversal T> for n = 4,
s = 1 and the dihedral group D4 . It is important to note that we had to check 39 configurations for
canonicity, out of 34 = 81 spin configurations, which is not bad. Moreover, we note that each of the
canonic representatives arises just once from an element of smaller total magnetization (total weight). The
corresponding generalization reads as follows:
4.7.3 Theorem (Read) Assume a finite action
GX
X=
where X is totally ordered by ≤ and a disjoint union
n
[
Xi ,
i=1
of invariant and nonempty subsets Xi . Let A denote an algorithm that produces for each x ∈ X either the
empty set or a set A(x) ⊆ X in descending order, such that the following conditions hold for the canonic
transversals T>i of the orbit sets G\\Xi , for each i ∈ n − 1:
• T>i+1 ⊆
S
A(x), union over the x ∈ T>i },
• for each x ∈ T>i+1 there exists a unique element x0 ∈ T>i such that x ∈ A(x0 ), and finally,
4.7. Orderly generation
141
• for any x1 , x2 ∈ T>i+1 , x1 < x2 , we have the implication
x1 ∈ A(x01 ), x2 ∈ A(x02 ) imply x01 < x02 .
Then, computing T>1 and recursively running through T>i with x, producing the augmentation A(x), and eliminating representatives which are not canonic from A(x), for i ∈ n − 1, gives the desired canonic transversal
T> of G\\X as the following union:
n
[
T> =
T>i .
i=1
2
This orderly generation obviously will work as soon as we have found an augmentation procedure A that
has the properties listed in the items of the above theorem of Read. We already saw such an augmentation
that can be used for the spin configurations and which amounts to an increase of magnetization by 1. It has
to be stressed that the augmentation strongly depends on the action G X and, obvioulsly, on both — G and
X. Therefore, this action has to be considered in details before we start dealing with transversals of G Y X .
The necessary canonicity test can be done quite efficiently using any suitable invariant that allows to put the
canonic representatives in what is called an AVL–tree in computer science. For example, in the molecular
graphs case, we can use a so-called Morgan table, which is a numbering well known to chemists, and used
in the coding procedure for chemical substances in the Chemical Abstracts. It reduces the set of checks to a
reasonable small number.
I am afraid that physicists do not read Chemical Abstracts and do not know the Morgan table (at least I do not
know both: the AVL–tree and the Morgan table). WSF
Summing up we should like to mention that an efficient way of evaluating a transversal of the symmetry of
G on Y X can be implemented by using the following methods:
• A recursion procedure, using recursion on |Y |.
• Read’s orderly generation, by applying lexicographic or any other ordering of Y X .
• A canonic form of the elements f ∈ Y X which allows to put the canonic representatives (canonic with
respect to the ordering mentioned in the second item) together into an AVL-tree and which reduces
the necessary number of checks to a reasonably small one.
file:f47.tex
142
4.8
Chapter 4. Classes of configurations, constructive aspects
Generating orbit representatives
The evaluation of an orbit transversal is of limited use since these sets are usually very big. We presented some
cardinalities for spin configurations on a cube in example 3.2.4. If we want to consider spin configurations
2
one a square lattice N × N then the number of configurations is (2s + 1)N , whereas the order of symmetry
group is |DN o S2 | = 8N 2 . Assuming that all orbits are asymmetric we obtain the lower limit of the number
2
of orbits as (2s + 1)N /8N 2 . Since the numerator increases much faster than the denominator then large
lattices lead to very big transversals. For example there almost 360000 orbits for N = 4 and s = 1. Even
restricting ourselves to orbits with given magnetization (weight, in general) we have to deal with big sets (in
the previous example there are 42967 orbits with M = 0). These numbers are even larger for graphs, since
considering graphs on v vertices we, in fact, investigate mappings determined on v2 and, for example, there
are approximately 109 graphs on 11 vertices. Thus, we are looking for a method which allows to examine
further cases, e.g. graphs of more than ten vertices, say. In order to cover such cases with some success, we
have to accept redundancy and hence we shall run into the isomorphism disease. Invariants, isomorphism
tests and all that will have to be discussed. To begin with we need to check that we can generate orbit
representatives uniformly at random.
821
4.8.1 The Dixon/Wilf Algorithm If G X denotes a finite action, then we can generate orbit representatives
uniformly at random in the following way:
• Choose a conjugacy class C of G with probability
p(C) :=
|C| |Xg |
, where g ∈ C.
|G| |G\\X|
• Pick any g ∈ C and generate a fixed point x of g, uniformly at random.
Then the probability that x is an element of the orbit ω ∈ G\\X is 1/|G\\X|, i.e. x is uniformly distributed
over the orbits of G on X.
The application of this method to the generation of representatives of G–classes, H–classes, H × G–classes
and H oX G–classes on Y X is easy, since we know the fixed points very well. Let us consider the G–classes,
for example.
822
4.8.2 Corollary For finite G X and Y the following procedure yields elements f ∈ Y X that are distributed
over the G–classes on Y X uniformly at random:
• Choose a conjugacy class C of G with the probability
0
|C| |Y |c(ḡ )
p(C) := P
,
c(ḡ)
g |Y |
g 0 ∈ C.
• Pick any g ∈ C, evaluate its cycle decomposition and construct an f ∈ Y X that takes values y ∈ Y on
these cycles which are distributed uniformly at random over Y .
Consider spin configurations on a cube with s = 1/2:
823
4.8.3 Example We would like to generate configurations of 8 spins s = 1/2 under the action of Oh . The
conjugacy classes of this group have orders (cf. ??) 1, 6, 1, 3, 3, 6, 8, 6, 6, and 8. The corresponding numbers
of cyclic factors of the permutations induced on the set 8 of cube vertices are 8, 6, 4, 4, 4, 4, 4, 2, 2, and 2,
so that the numbers of fixed points on the set {−1/2, 1/2}8 of spin configurations amount to 1 · 256, 6 · 64,
1 · 16, 3 · 16, 3 · 16, 6 · 16, 8 · 16, 6 · 4, 6 · 4, and 8 · 4. This yields for the probabilities p(C) the values
32 48 2
6
6 12 16 3
3
4
,
,
,
,
,
,
,
,
,
.
132 132 132 132 132 132 132 132 132 132
4.8. Generating orbit representatives
143
These numbers may be multiplied by the common denominator 132 in order to get the natural numbers,
which we accumulate obtaining
32, 80, 82, 88, 94, 106, 122, 125, 128, 132.
A generator that yields natural numbers between 1 and 132 uniformly at random is now used in order to
choose C. As soon as it generates one of the numbers 1,2,. . . ,32 this means that the first one of the conjugacy
classes is chosen. If it generates 33,34,. . . ,80, the second class is chosen, and so on. Assume that it generates
112, then the seventh conjugacy class is picked, an element of which is the permutation (1)(234)(5)(678). A
random generator of zeros and ones is now used in order to associate values −1/2 if 0 is generated or 1/2
otherwise with the cyclic factors of the chosen element. If it generates the sequence 1,1,0,1, say, we obtain
the spin configuration with −1/2 at the vertex 5 and 1/2 at the others, i.e.
t
t
t
d
t
t
t
t
3
This method can easily be refined in order to generate configurations with prescribed magnetization or,
more generally, representatives of orbits with a given weight, uniformly at random. If G X is a finite action,
w: X → R a weight function, then G acts on the union of orbits with a prescribed weight r ∈ R, i.e. on the
inverse image w−1 [{r}] of r, and we need only to replace in 4.8.1 the corresponding factors in the numerator
and in the denominator obtaining
4.8.4 Corollary If we choose the conjugacy class C ⊆ G with the probability
p(C) :=
|C| |w−1 [{r}]g |
,
|G| |G\\w−1 [{r}]|
82
g ∈ C,
pick any element g ∈ C and generate a fixed point of weight r of g uniformly at random, then we obtain an
element of X that is uniformly distributed over the orbits of weight r.
4.8.5 Example Let us again consider spin configurations on a cube for s = 1/2. As the weight we take ra
magnetization for s = 1/2.
We know from example 3.2.4 that for s = 1/2 there are six orbits with M = 0, so the denominator is equal
to 48 · 6 = 288. The first class contains only the identity element 1, so the number of configurations fixed by
this element and of weight M = 0 is equal to the total number of such configurations, i.e.
8
−1
|M [{0}]1 | =
= 70,
4
so this class should be chosen with the probability 35/144. The next class consits of 6 elements with the
cyclic structure (22 , 14 ), each them fixes 26 = 64 configurations. We have to find out how many of these
configurations have magnetization M = 0. Since fixed configurations are constant on cycles then denoting
by fi , 1 ≤ i ≤ 6, the value of f on elements of the i-th cycle we have
M = 2f1 + 2f2 + f3 + f4 + f5 + f6 .
The first two summands can give: 2 in one case, 0 in two cases, and –2 in one case. Therefore, since M = 0
then we have to find all possible values of f3 , . . . , f6 such that their sum is equal to –2, 0, and 2, respectively.
The first and the last results are possible only for f3 = . . . = f6 = ∓1/2, whereas we can obtain 0 in 42 = 6
ways. Summing up these results we obtain that each element g in the second class fixes 14 configurations
with M = 0, so this class will be chosen with the probability 6 · 14/288 = 7/24 = 42/144. In the same way
we can find the numerators for the other conjugacy classes and — after accumulating as in the previous
example – we have the following series:
35, 77, 80, 89, 98, 116, 132, 132, 144, 144
144
Chapter 4. Classes of configurations, constructive aspects
(it is correct, elements of the last class do not fix configurations with M = 0). If we obtain, for example,
again 112 from our generator then we pick the sixth class containig elements with the cyclic structures (24 )
(they are twofold rotations with axes parallel to face diagonals). If we choose the element (18)(26)(37)(45)
from this class and next generate the sequennce 1, 0, 1 (spin projections for the fourth cycle are determined
by the previous three), then we obtained the following configuration with M = 0 and fixed by an element of
the sixth class:
t
d
d
d
t
t
d
t
3
This example shows that it may be a quite cumbersome to find the number of fixed points for a given
g ∈ G and given weight, i.e. the factor |w−1 [{r}]g | in the numerator, especially when the cyclic structure of
elements in the picked conjugacy class is complicated — it is much easier to deal with elements having all
cyclic factors of the same length. Moreover, the number of conjugacy classes increases if we consider larger
groups. For example, in the case when the acting group is a symmetric group, the choice of a conjugacy
class amounts to the choice of a proper partition. We therefore should not forget that the number of proper
partitions of n ∈ N is rapidly increasing with n. Here are a few of these numbers:
n
10
20
40
60
100
no. of proper partitions
42
627
37338
≈ 106
≈ 2 · 108
This shows that it is worthwile to try to avoid that this amount of probabilities is stored throughout the
whole generating process. For practical purposes one can start the generation and evaluate probabilities
only if required. This means that we need to evaluate p(Ci ) only if the generated random number exceeds
Pi−1
j=1 p(Cj ). The efficiency of this revised method heavily depends on the numbering of the conjugacy classes.
The numbering should clearly be chosen in such a way that p(Ci ) ≥ p(Ci+1 ). Asymptotic considerations
show (cf. Oberschelp, Dixon/Wilf) that in case of graph generation it is not bad to number the conjugacy
classes of the symmetric group according to the lexicographic order of partitions.
e , we can do the following.
In order to generate representatives of the orbits of G on X with prescribed type U
As each such orbit ω ∈ G\\X does contain elements x which have U as stabilizer, we can cut off from XU
each element that has a bigger stabilizer. It clearly suffices to remove from XU the elements belonging to
the XV , where U is maximal in V . The remaining subset of XU will be indicated as follows:
[
X̂U := XU \
XV .
U max. V
e \\\X with X̂U is the orbit of an x ∈ ω, where Gx = U , under the normalizer
The intersection of ω ∈ U
subgroup NG (U ), this is easy to check. As Gx = U we can even consider the action of the factor group
NG (U )/U on this intersection ω ∩ X̂U . This proves
831
e of G on X.
4.8.6 Lemma Each transversal of (NG (U )/U )\\X̂U is a transversal of the orbits of type U
Moreover, if we want to apply probabilistic methods, we can use that the orbits of NG (U )/U on X̂U are all
of the same order |NG (U )/U |. Hence we obtain
832
4.8.7 Theorem (Laue) By picking elements from X̂U uniformly at random, we obtain elements that are
e of G on X.
uniformly distributed over the orbits of type U
4.8. Generating orbit representatives
145
For example, if U := {1}, we have that X̂U = X̂1 is equal to X minus the union of the fixed point sets
XV , taken over all the minimal subgroups V ≤ G. If in particular G := Cn , then the minimal subgroups
V are just the cyclic subgroups Cp ≤ Cn , where p is prime. The same holds for G := Sn , but in this
case there may be, of course, several such Cp . For the dihedral groups Dn we have also to consider all n
two-element subgroups generated by reflections π k σ (cf. 1.4.15). As NG (1) = G, we can, according to 4.8.7,
easily generate representatives of the orbits of maximal length |Ḡ|, uniformly at random.
I removed the example on prymitive polynomials, since its ,,parent” had been removed. I was going to put
here, for example, the longest orbits of spin configuration on a square, but the obtained results are rather
too simple. If it is necessary I can provide an example with spin configurations on a cube or on a square.
Exercises
E 4.8.1
Prove that, for conjugate g, g 0 ∈ G, a finite action
GX
and an orbit ω ∈ G\\X, we have
|Xg ∩ ω| = |Xg0 ∩ ω|.
E 4.8.2 Let λ |= n describe the content (the most general weight) and let α(ḡ) ` n be the cycle partition
of g ∈ G, where G acts on X and |X| = n. How many mappings f : X → Y are fixed by g and have the
content λ? (I do know the answer and I do not whether it is possible to solve this problem but I think that
yes.)
file:f48.tex
E