SUPERCHARACTERS OF UNITRIANGULAR GROUPS AND SET
PARTITION COMBINATORICS
CARLOS A. M. ANDRÉ
Abstract. The representation theory of the symmetric group is a fundamental
model in combinatorial representation theory because of its connections to partition
and tableaux combinatorics. It has become clear in recent years that the superrepresentation theory of the finite unitriangular group has a similarly rich combinatorial structure built on set partitions. The main purpose of this course is to introduce
the supercharacter theory of the unitriangular group and to explore its connections
with set partition combinatorics. In particular, we will describe the set partition
combinatorics appearing in connection to restriction, superinduction, inflation and
deflation of supercharacters, and show how these can be used to obtain a relationship
between the supercharacter theory of all unitriangular groups simultaneously and
the combinatorial Hopf algebra of symmetric functions in non-commuting variables
that mirrors the symmetric group’s relationship with the Hopf algebra of symmetric
functions.
Contents
Introduction
1. Review of character theory
1.1. Characters
1.2. Class functions
1.3. Restriction and induction
1.4. Characters of finite abelian groups
1.5. Supercharacter theories of finite groups
Exercises
2. Supercharacters and superclasses for algebra groups
2.1. Algebra groups
2.2. Superclasses
2.3. G-actions on the dual group A◦
2.4. Supercharacters
2.5. Orthogonality of supercharacters
2.6. Supercharacter values
2.7. Restriction, products and superinduction of supercharacters
Exercises
3. Supercharacters and superclasses of the unitriangular group
3.1. Set partitions
3.2. k× -coloured set partitions
2
7
7
9
9
11
12
13
15
15
16
17
17
19
20
23
28
30
30
32
Date: July 3, 2013.
2010 Mathematics Subject Classification. 20C15, 20C33, 16T30, 16T05, 05E05.
Ackowlegments: The author thanks the Organising Committee of ECOS2013 for the invitation
and opportunity to teach this course in the CIMPA school: “Modern Methods in Combinatorics
ECOS2013” (“2da Escuela Puntana de Combinatoria: Escuela de Combinatoria del Sur”, Universidad
Nacional de San Luis, Argentina, July 22-August 2, 2013).
1
2
CARLOS A. M. ANDRÉ
3.3. Superclasses of Un
3.4. Supercharacters
3.5. Supercharacter values
3.6. A uncoloured supercharacter theory
3.7. Non-crossing and non-nesting set partitions
Exercises
4. The Hopf algebra of superclass functions
4.1. Hopf algebra basics
4.2. Symmetric functions in non-commuting variables
4.3. Representation theoretic functors on SC
4.4. The Hopf algebra structure of SC
4.5. The subalgebra LSC
Exercises
References
33
34
36
39
40
42
44
44
46
50
57
61
63
64
Introduction
In representation theory, abstract algebraic structures are represented using matrices or geometry. These representations provide a bridge between the abstract symbolic
mathematics and its explicit applications in many branches of mathematics as well as in
related fields such as physics, chemistry, engineering, and statistics. In Combinatorial
Representation Theory, combinatorial objects are used to model these representations,
and are refined enough to help describe, count, enumerate, and understand the representation theory. Furthermore, the interplay between the algebra and the combinatorics
goes both ways: the combinatorics helps answer algebraic questions and the algebra
helps answer combinatorial questions.
The central object of representation theory is an algebra A over a field k (or simply
a k-algebra); that is, a vector space over k with a multiplication which is associative,
distributive, has an identity 1 = 1A , and satisfies (ra)b = a(rb) = r(ab) for all a, b ∈ A
and r ∈ k. An important example is the group algebra kG of a finite group G. Formally,
kG is a vector space over k having G as a k-basis where the multiplication is inherited
from the multiplication in the group.
The main goal of representation theory is to study a given algebra A via its actions
on k-vector spaces. An (left) A-module is a finite-dimensional vector space M over k
with an A-action A × M → M, (a, m) 7→ am, which satisfies 1u = u, a(bu) = (ab)u,
(a + b)u = au + bu and a(ru + sv) = r(au) + s(av) for all a, b ∈ A, u, v ∈ M and
r, s ∈ k. We sometimes use the words “module” and “representation” interchangeably.
Representation theorists are always trying to break up modules into pieces. An Amodule M is said to be:
• indecomposable if M is not the direct sum of two nonzero submodules;
• irreducible or simple if the only submodules of M are the zero module 0 and
M itself;
• semisimple if it is the direct sum of simple submodules.
A k-algebra is semisimple if all A-modules are semisimple; in fact, A is semisimple
if every indecomposable A-module is irreducible (the converse is always true). As an
example, if G is a finite group and k is a field of characteristic zero (for example, if
k = C is the field of complex numbers), then the group algebra kG is semisimple.
SUPERCHARACTERS AND SET PARTITION COMBINATORICS
3
Some of the fundamental questions of representation theory are to find the simple
A-modules, to index/count them, to determine their dimensions, and to evaluate their
characters. The character of an A-module M is the function χM : A → k where
χM (a) = Tr(a, M), for a ∈ A, is the trace of the linear transformation determined by
the action of a on M; more precisely,
X
χM (a) =
ai,i
1≤i≤m
where (u1 , . . . , um ) is a k-basis of M and
X
auj =
ai,j ui ,
1 ≤ j ≤ m.
1≤i≤m
Special modules often have particularly nice formulas describing their characters. It
is important to note that having a nice character formula for M does not necessarily
mean that it is easy to see how M decomposes into irreducibles. Thus the question of
decomposing a module into irreducibles is really different from the question of knowing
its character. Finally, we mention that a representation may be particularly interesting
just because of its structure while other times it is a special representation that helps
to prove some particularly elusive theorem. Sometimes these special representations
lead to a completely new understanding of previously known facts.
The adjective “Combinatorial” in Combinatorial Representation Theory refers to the
way in which we give the answers to the main questions of representation theory: we
want to parametrise the irreducible modules by “nice combinatorial objects”, to have
a formula for the dimensions that counts the
Pnumber of “nice combinatorial objects”,
or a character formula of the type χ(a) = T wta (T) where the sum runs over a set
of “nice combinatorial objects” and wta is a weight on these objects depending on the
element a where we are evaluating the character. On the other hand, we want to give
constructions that have a very explicit and very combinatorial flavour. It is particularly
pleasing when interesting representations arise in other parts of combinatorics, and
sometimes a representation is exactly what is most helpful for solving a particular
combinatorial problem. The main point of all this is that a combinatorialist thinks
in a special way (nice objects, bijections, weighted objects, etc.), and this method of
thinking should be an integral part of the form of the solution to the problem.
A prototype example is the representation theory of the symmetric group Sn consisting of all permutations of the set [n] = {1, 2, . . . , n}; as for every finite group, by
a representation of Sn we mean a representation of its group algebra A = CSn . It is
well-known that the irreducible CSn -modules are in one-to-one correspondence with
integer partitions of n; by an integer partition of n we mean a weakly decreasing sequence λ = (λ1 , . . . , λℓ ) of positive integers, called parts, summing to n; in this case,
we write λ ⊢ n, and denote the number of parts, or length of λ, by ℓ = ℓ(λ). It is
standard to identify a partition λ ⊢ n with its Ferrers diagram which has λi boxes in
the ith row. For example, λ = (4, 2, 2, 1) ⊢ 9 has Ferrers diagram
A standard tableau of shape λ is a filling of the Ferrers diagram of λ with the integers
1, 2, . . . , n such that the rows and columns are strictly increasing from left to right and
4
CARLOS A. M. ANDRÉ
from top to bottom respectively. For example,
1 2 4 9
3 6
5 8
7
is a standard tableaux of shape λ = (4, 2, 2, 1). If Sλ denotes the simple CSn -module
corresponding to the partition λ, then dimC Sλ equals the number of standard tableau
of shape λ. Finally, if χλ denotes the character of the simple CSn -module Sλ then its
value χλ (µ) at a permutation of cycle type µ = (µ1 , . . . , µs ), is given by
X
χλ (µ) =
wtµ (T)
T
where the sum is over all standard tableaux T of shape λ and
Y
wtµ (T) =
fi (T)
1≤i≤n
where


/ B(µ), and i + 1 is sw of i,
−1, if i ∈
fi (T) = 0,
if i, i + 1 ∈
/ B(µ), i + 1 is ne of i, and i + 2 is sw of i + 1,

1,
otherwise,
and B(µ) = µ1 + · · · + µk : 1 ≤ k ≤ s ; in this formula, “sw” means “strictly south”
and “weakly west”, and “ne” means “strictly north” and “weakly east”. In particular, we
see that every irreducible character of Sn is rational-valued, and hence it is a function
from Sn to the field Q of rational numbers.
In some favourable situations, it is possible to encode the representation theory of a
certain algebra inside some distinct algebraic structure which may have an interesting
combinatorial flavour. To illustrate this idea, we continue with the prototype
example
of the symmetric group Sn , let Rn be the Q-vector space with basis χλ : λ ⊢ n ;
thus, Rn consists of all rational-valued class functions defined on Sn , that is, functions
Sn → Q which are constant on the conjugacy classes of Sn . We may even be more
ambitious and aim to encode the representation theory of all symmetric groups Sn
simultaneously in a unique algebraic structure; one way of doing this is to form the
graded Q-vector space
M
Rn
R=
n∈N0
where by convention we agree that S0 = 1, so that R0 = Q; as it is usual, we write N to
denote the set of all positive integers, and set N0 = N ∪ {0} for the set of nonnegative
integers. Hence, R has a Q-basis consisting of all characters χλ where λ is any integer
partition (that is, a nondecreasing sequence λ = (λ1 , . . . , λm ) of positive integers).
The vector space R becomes an associative Q-algebra if we define a multiplication
R × R → R as follows. For all m, n ∈ N, the direct product Sm × Sn can be embedded
in a canonical way in the larger symmetric group Sm+n with Sm acting on the first m
integers, and Sn on the last n integers. If φ ∈ Rm and ψ ∈ Rn , then the mapping
(σ, τ ) 7→ φ(σ)ψ(τ ) defines a class function of Sm × Sn which we denote by φ × ψ, and
we define the product φ · ψ ∈ Rm+n to be the induced character
S
m+n
φ · ψ = IndSm
×Sn (φ × ψ)
SUPERCHARACTERS AND SET PARTITION COMBINATORICS
5
(for the basic notions of character theory see Section 1). This product makes R a
commutative graded Q-algebra where the adjective “graded” means that R0 = Q · 1
and Rm · Rn ⊆ Rm+n for all m, n ∈ N0 . It can be proved that R to be isomorphic (as a
graded algebra) to the graded Q-algebra Sym(X) consisting of all symmetric functions
on a countable infinite set X = {x1 , x2 , x3 , . . .} of commuting variables.
We recall the definition of Sym(X). Let Q[[X]] be the Q-algebra of formal power
series in the set X (with coefficients in the rational field). There is an action of the
elements σ ∈ Sn on this algebra given by the rule
(σf )(x1 , x2 , ...) = f (xσ(1) , xσ(2) , . . .)
where σ(i) = i for i > n. We say that f ∈ Q[[X]] is symmetric if it is invariant under the
action of Sn for all n ≥ 1. The Q-algebra of symmetric functions Sym(X) consists of all
symmetric functions f ∈ Q[[X]] which are also of bounded degree. This algebra has a
long history and is of interest in combinatorics, algebraic geometry, and representation
theory. For more information in this regard, see the texts of Fulton [22], MacDonald
[34], Sagan [37], or Stanley [43]. The algebra Sym(X) has several distinguished basis;
an very important example is the basis consisting of Schur symmetric functions. To
each partition λ = (λ1 , . . . , λm ) of a positive integer n ∈ N, we associate the function
sλ ∈ Q[[X]] which is defined as follows. A semistandard tableau of shape λ is a filling of
the Ferrers diagram of λ with positive integers such that the rows are weakly increasing
from left to right and the columns are strictly increasing from top to bottom. To
illustrate,
1 2 2 3
3 3
4 5
8
is a semistandard tableaux of shape λ = (4, 2, 2, 1). For every semistandard tableau
T of shape λ and every positive integer i, we let νi be the number of i’s appearing in
T, and define the monomial mT = xν11 xν22 xν33 · · · . Then, we define the Schur symmetric
function to be
X
sλ =
mT
T
where the sum is over all semistandard tableau of shape λ. It can be proved that indeed
sλ ∈ Sym(X), and that sλ : λ ⊢ n, n ∈ N0 is a Q-basis of Sym(X); we agree that
s∅ = 1 where ∅ is considered as the unique integer partition of 0. Furthermore, the
mapping χλ 7→ sλ defines an isomorphism of Q-algebras Φ : R → Sym(X); in particular,
we have Φ(χλ ·χµ ) = sλ sµ which means that the decomposition of the induced character
P ν
Sm+n
λ
µ
IndSm
×Sn (χ × χ ) can be determined from the Q-linear combination sλ sµ =
ν cλ,µ sν
where cνλ,µ ∈ Q, and vice-versa. This important result illustrates how the interplay
between the character theory of the symmetric groups and symmetric functions (on
commuting variables) has enriched both theories with very interesting combinatorics.
In fact, the space of symmetric functions has several algebraic operations (in particular
it is a Hopf algebra) and many interesting bases (Schur, power-sum, monomial, and
homogeneous symmetric functions). These algebraic operations and bases can be lifted
to the characters of the symmetric groups, and as such are meaningful representation
theoretic operations and bases. By the way of example, we mention also that the wellknown Frobenius Character Formula asserts that the character table of the symmetric
group is the transition matrix between the Schur basis and the power-sum basis (see
for example [22, Corollary 7.4]).
6
CARLOS A. M. ANDRÉ
In these lectures, we discuss a similar theory where the symmetric groups Sn are
replaced by the unitriangular groups Un (k) consisting of all upper triangular n × n
matrices with coefficients in a finite field k and ones on the diagonal. In order to achieve
this, we need to replace irreducible characters by supercharacters and conjugacy classes
by superclasses; in fact, the description of the irreducible characters, or the conjugacy
classes, of Un (k) is known to be a wild problem, and hence very hard to understand.
The notion of a supercharacter theory of a finite group was introduced by Diaconis and
Isaacs [18] to generalise an approach used by André (e.g. [5, 7, 8]) and Yan [45] to study
the irreducible characters of the finite groups Un (k). The basic idea is to coarsen the
usual irreducible character theory of a group by replacing irreducible characters with
linear combinations of irreducible characters that are constant on a set of clumped
conjugacy classes. This construction will be described in Section 2 for a larger family
of algebras groups and particularised to the unitriangular groups Un (k) in Section 3.
Supercharacters (and superclasses) of Un (k) have a nice parametrisation by coloured
set partitions of the set [n] where the colours are given by the nonzero elements of the
field k. We define a set partition π of [n] to be a family of nonempty sets, called blocks,
whose disjoint union is [n]; we write π = B1 /B2 / . . . /Bℓ ⊢ [n] where B1 , B2 , . . . , Bℓ
are the blocks of π which we agree to be ordered by increasing value of the smallest
element in the block (this implied order will allow us to reference the ith block of the
set partition without ambiguity). For instance, π = 146/25/3 is a set partition of
[6] = {1, 2, 3, 4, 5, 6} with parts {1, 4, 6}, {2, 5} and {3}. If π ⊢ [n] and 1 ≤ i < j ≤ n,
then the pair (i, j) is said to be an arc of π if i and j occur in the same block B of π and
there is no k ∈ B with i < k < j; we denote by D(π) the set consisting of all arcs of π.
Then, by a k× -colouring of π we mean a map φ : D(π) → k× where k× stands for the
set of all nonzero elements of k, and a k× -coloured set partition of [n] is a pair (π, φ)
consisting of a set partition π ⊢ [n] and a k× -colouring φ of π. Then, there is a oneto-one correspondence between k× -coloured set partitions of [n] and supercharacters of
Un (k); we write χπ,φ to denote the supercharacter associated with the k× -coloured set
partition (π, φ). Similarly, there is a one-to-one correspondence (π, φ) 7→ Kπ,φ between
k× -coloured set partitions of [n] and superclasses of Un (k). This correspondence is
established in Section 3 where we also prove that supercharacters take a constant value
on each superclass. Thus, it is possible to construct the supercharacter table of Un (k)
as the matrix with rows and columns indexed by k× -coloured set partitions where the
π,φ
entries are the values χπ,φ
σ,ψ of the supercharacter χσ,ψ on the superclass Kσ,ψ . As in
the case of the usual character table, the supercharacter table has orthogonal rows and
columns with respect to the usual Frobenius inner product of complex-valued functions
on Un (k); however, they cannot be orthonormal. In particular, we can define for each
n ∈ N the complex vector space SCn (k) consisting of all superclass functions defined
on Un (k) (that is, functions that are constant on the superclasses), and prove that the
supercharacters form an orthogonal basis of this space. Then, we can form the graded
vector space
M
SCn (k)
SC(k) =
n∈N0
where we agree that SC0 (k) = 1, so that SC0 (k) = C. Hence, SCn (k) has an orthogonal C-basis consisting of all supercharacters χπ,φ where (π, φ) is any k× -coloured set
partition. As in the case of the symmetric groups, it is possible to define a product
on the vector space SC(k) so that it becomes a graded C-algebra which turns out to
SUPERCHARACTERS AND SET PARTITION COMBINATORICS
7
be isomorphic (as a graded algebra) to a certain k× -coloured version of the graded Calgebra NCSym(X) consisting of all symmetric functions on a countable infinite set X =
{x1 , x2 , x3 , . . .} of non-commuting variables. In these lectures, we will use a simplified
uncoloured version of the algebra SC(k) by changing slightly the supercharacter theory
of Un (k); we refer to [1, Section 3.2] for the definition of this isomorphism in the general
coloured situation.
Most finite groups have more than one possible supercharacter theory, so there is
not really a canonical choice. For the unitriangular groups Un (k), there is a natural
coarsening of the classical supercharacter theory that is integer valued, and thus its
use may be combinatorially more natural and advantageous. In this supercharacter
theory, supercharacters and superclasses of Un (k) are parametrised by set partitions of
[n]; for every π ⊢ [n], we just define
X
[
χπ =
χπ,φ and Kπ =
Kπ,φ
φ
φ
where the sum and the union is over all k× -colourings of π. For example, in this
situation, the supercharacter table has a nice decomposition as a lower triangular
matrix times an upper triangular matrix (see [14]), and the corresponding graded
algebra SC(k) is always isomorphic to NCSym(X); in [1] it was required that k is the
field with 2 elements. By analogy with the symmetric group case, it is tempting to
define the product on SC(k) using induction of supercharacters, but this analogy is
misleading because in general the induction functor does not send superclass functions
to superclass functions. However, this is true for restriction functor; the supercharacter
theories of the unitriangular groups Un (k) as n varies behave nicely in the sense that
the restriction of a superclass function is a superclass function of the subgroup. To deal
with this obstruction we must use superinduction which in a certain sense is adjoint to
restriction at the level of superclass functions. Another obstacle is that restriction of
supercharacters depends on the choice of the embedding of a unitriangular group into
another. Indeed, each subset J ⊆ [n] determines a subgroup UJ (k) of Un (k) which is
naturally isomorphic to Um (k) for m = |J|; this subgroup is obtained by taking only
the rows and columns indexed by J. It turns out that “restriction” to the subgroup
Um (k) depends on the choice of the subset J ⊆ [n] with |J| = m. For this reason,
the definition of the operations on SC(k) must take these distinctions into account by
simultaneously considering all possible subgroups UJ (k). Then, the algebra structure
on SC(k) (indeed, the Hopf algebra structure) is obtained by following the natural
isomorphisms UJ (k) ∼
= Um (k) for m = |J|. The details will be discussed in Section 4.
1. Review of character theory
1.1. Characters. A C-linear representation of a finite group G is a homomorphism
of groups φ : G → GL(V ) where V is a finite-dimensional vector space over C. We
can extend φ linearly to obtain a homomorphism of C-algebras φ : CG → EndC (V )
where CG is the group algebra of G over C, whose elements are all formal C-linear
combinations
X
κg g, κg ∈ C,
g∈G
and where the multiplication is naturally defined in terms of the group multiplication
of G. Then, we can make CG acts on V by the rule κv = φ(κ)(v) (κ ∈ CG, v ∈ V ),
8
CARLOS A. M. ANDRÉ
and thereby we get a CG-module which we denote by (V, π), or simply by V 1. If V
and W are arbitrary CG-modules, then a CG-homomorphism φ : V → W is a C-linear
map satisfying φ(gv) = gφ(v) for all g ∈ G and all v ∈ V ; as usual, we denote by
HomCG (V, W ) the C-vector space consisting of all CG-homomorphisms from V to W ,
and set EndCG (V ) = HomCG (V, V ). (Then, the (complex) representation theory of G
could be defined as the study of the category whose objects are the CG-modules and
whose the morphisms are just the CG-homomorphisms.)
Given any CG-module V = (V, φ), the character of G afforded by V is defined to be
the map χV : G → k where χV (g) = Tr(g, V ) is the trace of the linear transformation
π(g) ∈ GL(V ) determined by g ∈ G; thus, if n = dim V and {v1 , . . . , vn } is a basis of
V , then
X
χV (g) =
ci,i (g)
1≤i≤n
where the functions ci,j : G → k for 1 ≤ i, j ≤ n are defined by the rule
X
gvj =
ci,j (g)vi
1≤j≤n
for all g ∈ G. A function χ : G → C is called a (complex) character of G if χ = χV
is the character of G afforded by some CG-module V . The value χ(1) is called the
degree of χ, and it clearly equals the dimension of every CG-module which affords χ.
Characters of degree 1 are called linear characters, and they correspond exactly to the
group homomorphisms from G to the multiplicative group C× of C. In particular, the
function 1G with constant value 1 on G is a linear character, to which we refer as the
principal, trivial, or unit character of G.
By Maschke’s Theorem the group algebra CG is semisimple which means that as
a CG-module (for the natural action given by left multiplication) the group algebra
CG decomposes as a direct sum of irreducible CG-submodules; recall that a CGmodule V is irreducible if {0} and V are the only CG-submodules of V . In fact,
Maschke’s Theorem asserts that every CG-module decomposes as a direct sum of irreducible CG-modules. In particular, if we choose a representative set of irreducible
CG-modules S1 , . . . , Sm affording characters χ1 , . . . , χt respectively, then we obtain the
set irr(G) = {χ1 , . . . , χt } of all irreducible characters of G (that is, characters afforded
by irreducible CG-modules). Moreover, it also follows that every character χ of G is a
linear combination
χ = m1 χ1 + m2 χ2 + · · · + mt χt
where m1 , . . . , mt ∈ N0 , not all equal to zero. In particular, it is well-known from the
theory of semisimple algebras that the character ρG afforded by CG (as a CG-module)
decomposes as the sum
X
ρG =
χ(1)χ;
χ∈irr(G)
we refer to ρG as the regular character of G (and to CG as the regular CG-module).
It is easy to see that ρG (g) = |G|δg,1 for all g ∈ G, and thus we deduce the important
formula
X
(1a)
|G| =
χ(1)2 .
χ∈irr(G)
1We
may define right CG-modules in an obvious and entirely similar way. However, otherwise
stated, the term “CG-module” will always refer to a “left CG-module”.
SUPERCHARACTERS AND SET PARTITION COMBINATORICS
9
1.2. Class functions. Every character of G clearly takes a constant on each conjugacy
class of G; recall that the conjugacy class of G which contains the element g ∈ G is
the set h−1 gh : h ∈ G which we denote by Cg . Two conjugacy classes of G are either
equal of disjoint, and G is the disjoint union of all its conjugacy classes; if C is a
conjugacy class of G, then we write κC to denote the characteristic function of C, that
is, the function κC : G → C where for any g ∈ G the value κC (g) is equal to 1 or 0
according to g ∈ C or not. A function ϑ : G → C is said to be a class function of G (or
a central function) if it takes a constant value on each conjugacy class; equivalently,
if ϑ(gh) = ϑ(hg) for all g, h ∈ G. The set consisting of all class functions of G forms
a vector space over C which we denote
if cl(G) denotes the set
by cf(G). Obviously,
of all conjugacy classes of G, then κC : C ∈ cl(G) is a C-basis of cf(G), and hence
dimC cf(G) = |cl(G)|; the cardinality |cl(G)| is called the class number of G and is
usually denoted by kG . Since every character of G is a class function, we conclude that
|irr(G)| ≤ kG ; indeed, although not trivially, it can be proved that irr(G) is also a
C-basis of cf(G), and thus |irr(G)| = kG . This basis irr(G) has nice properties; in
particular, it is orthonormal with respect to the standard scalar product of complexvalued functions
onG. For any pair of functions ξ, ζ : G → C, we define the Frobenius
scalar product ξ, ζ by the formula
1 X
(1b)
ξ, ζ =
ξ(g)ζ(g);
|G| g∈G
we write ξ, φ G whenever it is necessary to emphasise that the Frobenius product is
taken for functions on G.
The first orthogonality relation (see [26, Corollary 2.14]) asserts that
χ, χ′ = δχ,χ′
for all χ, χ′ ∈ irr(G), and
it is not hard to conclude that a character χ of G is
irreducible if and only if χ, χ = 1. This is essentially the content of Schur’s lemma
([26, Lemma 1.5]) which asserts that, if V and W are irreducible CG-modules, then
every nonzero CG-homomorphism φ ∈ HomCG (V, W ) has an inverse in HomCG (W, V ).
In particular, we deduce that EndCG (V ) = C · idV for all irreducible CG-module V ;
here, we write idV to denote the identity map of V . In general, if V and W are arbitrary
CG-modules affording characters χV and χW respectively, then
(1c)
χV , χW = dimC HomCG (V, W ).
L
L
In fact, if V = i∈I Si and W = j∈J Sj are decompositions of V and W as direct
sums of irreducible CG-submodules, then there is an C-linear isomorphism
M
HomCG (Si , Sj ),
HomCG (V, W ) ∼
=
(i,j)∈I×J
and hence the result is an immediate consequence of Schur’s lemma. Finally, we observe
that from this it is easily deduced that two CG-modules are CG-isomorphic if and only
if they afford the same character.
1.3. Restriction and induction. We now look at ways of relating the representations
of the group G to the representations of its subgroups. Firstly, we introduce the
elementary idea of restricting a CG-module V to a subgroup H of G. Since the group
algebra CH is a subalgebra of CG, the vector space V is also a CH-module which we
denote by ResG
H (V ). The character of H afforded by VH is of course given by restricting
to H the character χ = χV afforded by V ; we denote this restriction by ResG
H (χ). More
10
CARLOS A. M. ANDRÉ
generally, if denote by ResG
H (φ) the restriction to H of any map φ : G → C, then the
G
mapping φ 7→ ResH (φ) defines a C-linear map ResG
H : cf(G) → cf(H).
Less trivial are the notions of induced module and induced character. First of all,
for every class function φ ∈ cf(H), we write φ◦ to denote the extension by zero of φ to
G (that is, φ◦ (h) = φ(h) if h ∈ H, and φ◦ (g) = 0 if g ∈ G \ H), and define the induced
class function φG ∈ cf(G) by the formula
1 X ◦
(1d)
φG (g) =
φ (xgx−1 )
|H| x∈G
for all g ∈ G; notice that φG (1) = |G|/|H| = |G : H|. For our purposes, it is
G
convenient to use the alternative notation IndG
H (φ) instead of φ ; hence, we obtain
G
a map IndH : cf(H) → cf(G) which is easily seen to be C-linear and adjoint to
ResG
H : cf(G) → cf(H) in the sense that
G
(φ)
=
φ,
Res
(ψ)
(1e)
ψ, IndG
H
H
G
H
for all ψ ∈ cf(G) and all φ ∈ cf(H); this equality is known as Frobenius reciprocity.
It is also straightforward to check that, if φ is a character of H, then IndG
H (φ) is a
character of G; in particular, it is afforded by some CG-module (which is expected to
be related to the CH-module affording φ). In fact, since the group algebra CG is a right
CH-module (where the action of H is given by right multiplication), we can construct
the tensor product CG ⊗CH W for every (left) CH-module W . This tensor product
becomes a CG-module with respect to the action given by g · (g ′ ⊗ w) = (gg ′) ⊗ w for
all g, g ′ ∈ G and all w ∈ W , and it is not hard to prove that it affords the induced
G
character IndG
H (χW ); we thus define the induced CG-module IndH (W ) to be the CGmodule CG ⊗CH W .
Induced modules are characterised by the existence of a imprimitivity decomposition;
in other words, a CG-module V is the induced CG-module IndG
H (W ) for some subgroup
H of G and some CH-module W if and only if V decomposes as a direct sum V =
W1 ⊕ · · · ⊕ Wt of C-vector subspaces W1 , . . . , Wt which are transitively permuted by G
(that is, for every 1 ≤ i, j ≤ t, there exists g ∈ G such that gWi = Wj ). In fact, if we
set W = W1 and define H to be the stabiliser
H = StabG (W ) = g ∈ H : gW = W
G
of W in G, then W is a CH-module such that
L V = IndH (W ) (for a proof see [26,
Theorem 5.7]); moreover, it follows that V = t∈T tW where T ⊆ G is a complete set
of representatives for the left cosets of H in G.
A special situation occurs when the CG-module V has a C-basis vω : ω ∈ Ω
indexed by some finite set Ω where G acts as a group of permutations in such a way
that gvω = vg·ω for all g ∈ G and all ω ∈ Ω; here, we write g · ω to denote the action of
g ∈ G on ω ∈ Ω. In this case, we say that V is a permutation CG-module, and refer to
the character χV afforded by V as the permutation character of G on Ω. It is an easy
exercise to prove that
χV (g) = ω ∈ Ω : g · ω = ω for all g ∈ G; furthermore, if G acts transitively on Ω and ω ∈ Ω is arbitrary, then
χV = IndG
of the subgroup H = StabG (ω).
H (1H ) is induced by the principal
character
In the general situation, the multiplicity χV , 1G G is precisely the number of orbits of
G on Ω; in other words, this means that if χ is the permutation character of G on Ω
SUPERCHARACTERS AND SET PARTITION COMBINATORICS
11
and t is the number of orbits of G on Ω, then
1 X
(1f)
t = χ, 1G G =
χ(g)
|G| g∈G
(this formula is known as the orbit-counting formula and is erroneously attributed to
Burnside).
1.4. Characters of finite abelian groups. In this subsection, we consider the character theory of a finite abelian group A where the operation will be written additively. Since A is abelian, every conjugacy class of A consists of a unique element,
and thus A has exactly |A| conjugacy classes. Therefore, we have |irr(A)| = |A|, and
hence (1a) implies that every irreducible character of A is linear (that is, has degree
1). Let ϑ ∈ irr(A) be arbitrary. Being the trace of a one-dimensional representation of A, we see that ϑ(a + b) = ϑ(a)ϑ(b) for all a, b ∈ A. In particular, we have
1 = ϑ(0) = ϑ(a − a) = ϑ(a)ϑ(−a), and hence ϑ(a) 6= 0 and ϑ(a)−1 = ϑ(−a) for all
a ∈ A. It follows that ϑ is a group homomorphism ϑ : A → C× . On the other hand, if
ϑ, ϑ′ ∈ irr(A), then it is easy to see that ϑϑ′ ∈ irr(A) where (ϑϑ′ )(a) = ϑ(a)ϑ′ (a) for
all a ∈ A 2. It follows that irr(A) is a group with respect to the multiplication of characters; we write A◦ to denote this group, and refer to A◦ as the dual group of A 3. We
note that the inverse ϑ−1 of a character ϑ ∈ A◦ is given by ϑ−1 (a) = ϑ(a)−1 = ϑ(−a)
for all a ∈ A; in fact, since every element a ∈ A has finite order, we see that ϑ(a) is a
root of unity, and so
ϑ(−a) = ϑ(a)−1 = ϑ(a)
for all a ∈ A 4.
Now, suppose that B is a subgroup of A, and consider the dual group B ◦ of B. In this
situation, the restriction of characters clearly defines a surjective group homomorphism
◦
◦
◦
ResA
B : A → B with kernel consisting of all linear characters ϑ ∈ A such that ϑ(b) = 1
for all b ∈ B (that is, such that B ⊆ ker(ϑ) where we write ker(ϑ) to denote the kernel
of ϑ : A → C). Therefore, if we define the orthogonal subgroup B ⊥ to be the subgroup
B ⊥ = ϑ ∈ A◦ : B ⊆ ker(ϑ)
of A◦ , then we obtain a group isomorphism B ◦ ∼
= A◦ /B ⊥ , and in particular we see that
⊥
◦
◦
|B | = |A |/|B | = |A|/|B|.
On the other hand, we consider induction of characters from the subgroup B to A.
Let σ ∈ B ◦ , and let ϑ ∈ A◦ be any irreducible constituent of the induced character
IndA
B (σ). Then, by Frobenius reciprocity, we have
A
0 6= ϑ, IndA
B (σ) A = ResB (ϑ), σ B .
A
Since ResA
B (ϑ) and σ are irreducible characters, we conclude that ResB (ϑ) = σ (that is,
A
A
ϑ is an extension of σ to A). In fact, we have ResB (ϑτ ) = ResB (ϑ) = σ for all τ ∈ B ⊥ ,
and thus ϑτ is also an irreducible constituent of IndA
B (σ). Since the mapping τ 7→ ϑτ
2More
generally, if G is an arbitrary finite group, then the product χχ′ of two characters χ and χ′
of G is always a character of G; for a proof, see [26, Theorem 4.1].
3We note that |A◦ | = |A| and that for every a ∈ A the mapping ϑ 7→ ϑ(a) defines a linear character
of A◦ ; moreover, it is easily seen that the mapping a 7→ ϑ 7→ ϑ(a) defines a group isomorphism
A∼
= (A◦ )◦ .
4In fact, for every finite group G and every character χ of G, it is true that χ(g −1 ) = χ(g) for all
g ∈ G; see [26, Lemma 2.15].
12
CARLOS A. M. ANDRÉ
⊥
defines a bijection A◦ → A◦ , we see that IndA
B (σ) has at least |B | = |A|/|B| = |A : B|
A
irreducible constituents. Since IndB (σ)(0) = |A : B|σ(0) = |A : B|, it follows that
X
X
(1g)
IndA
(σ)
=
ϑτ
=
ϑ
·
τ.
B
τ ∈B ⊥
notice that IndA
B (1B ) =
P
τ ∈B ⊥
τ ∈B ⊥
A
τ , and thus IndA
B (σ) = ϑ · IndB (1B ).
1.5. Supercharacter theories of finite groups. A supercharacter theory of a finite
group G is a pair (X , Y) where X is set partition of G, and Y an orthogonal set of
characters of G (not necessarily irreducible), such that:
(1) |X | = |Y|,
(2) every character χ ∈ Y takes a constant value on each member K ∈ X , and
(3) each irreducible character is a constituent of one of the characters χ ∈ Y.
We refer to the members K ∈ X as superclasses and to the characters χ ∈ Y as
supercharacters of G; once a supercharacter theory for G is fixed, we will also use the
notation scl(G) for the set of superclasses and sch(G) for the set of supercharacters.
We note that the superclasses of G are always unions of conjugacy classes; moreover,
{1} is always a superclass and the principal character 1G is always a supercharacter.
For an arbitrary finite group G, there are two “trivial” supercharacter theories: in
one, Y = irr(G) consists just of all irreducible characters of G, and in the other
Y = {1G , ρG − 1G }. The corresponding superclasses in the first case are just the
conjugacy classes of G, and in the second case, they are the sets {1} and G \ {1}.
Although for some groups these trivial examples are the only possibilities, there are
many groups for which nontrivial supercharacter theories exist. For example, suppose
that A is a group that acts via automorphisms on the given group G; in other words,
there is a group homomorphism φ : A → Aut(G) where Aut(G) denotes the group
consisting of all automorphisms of G. Then, A acts on the set cl(G) conjugacy classes
of G; if a ∈ A and
C ∈ cl(G) is the conjugacy class which contains g ∈ G, then
aC = ah : h ∈ C is the conjugacy class which contains ag = φ(a)(g). We define
the members of X to be the unions of the A-orbits on cl(G). On the other hand, A
also permutes both the irreducible characters of G; if a ∈ A and χ ∈ irr(G), then
aχ ∈ irr(G) is defined by (aχ)(g) = χ(a−1 g) for all g ∈ G. We then define for every
A-orbit Ω ⊆ irr(G) the character
X
χΩ =
χ(1)χ,
χ∈Ω
and let Y consist of all such characters. Since irreducible characters are orthonormal,
it is clear that Y is an orthogonal set of characters. Axioms (2) and (3) are also
clear from the definitions, and hence the hard part is to justify that A has the same
number of orbits on irr(G) and on cl(G). This follows by a result of Brauer (see [26,
Theorem 6.32] which asserts that for every a ∈ A the number of elements fixed by a
on cl(G) equals the number of elements fixed by a on irr(G). Then, the counting
formula (1f) implies that
1 X
|X | =
ϑ(a) = |Y|
|A| a∈A
where ϑ(a) is the number of elements fixed by a ∈ A on cl(G).
SUPERCHARACTERS AND SET PARTITION COMBINATORICS
13
Exercises.
1.1. ([18, Lemma 2.1]) Let G be a finite group, X a partition of G, and Y an orthogonal
set of characters of G. Assume that |X | = |Y| and that the characters ξ ∈ Y are
constant on the sets X ∈ X . Prove that the following are equivalent:
(1) The set {1} is a member of X .
P
(2) For every ξ ∈ Y, there is c(ξ) ∈ C such that c(ξ)ξ = χ∈I(ξ) χ(1)χ.
(3) Each irreducible character χ ∈ irr(G) is a constituent of one of the characters
ξ ∈ Y.
1.2. Let p be a prime number, let Fq denote the finite field with q = pe (e ≥ 1)
e
elements, and define the trace map Tr : Fq → Fp by Tr(α) = α + αp + · · · + αp −1 for
α ∈ Fq ; notice that Fp can be identified with Z/Zp = {0, 1, . . . , p − 1}, and thus for
every α ∈ Fq we may think of Tr(α) as an integer modulo p.
(1) For every α ∈ Fq , define ϑα : Fq → C× by
ϑα (β) = exp 2πi Tr(αβ)
p
for all β ∈ Fq . Prove that irr(Fq+ ) = ϑα : α ∈ Fq where Fq+ denotes the
additive group of the field Fq .
(2) More generally, for any n ∈ N, consider the additive group A = (Fq+ )n , and
define x · y = x1 y1 + · · · + xn yn where x = (x1 , . . . , xn ) and y = (y1 , . . . , yn ) are
elements of A. For every x ∈ A, define ϑx : A → C× by
ϑx (y) = exp 2πi Tr(x·y)
p
for all y ∈ A. Prove that irr(A) = ϑx : x ∈ A .
1.3. ([16, Section 2]) Let k be a finite field, and consider the abelian group A = kn
(with respect to addition). For every x, y ∈ A, define x · y as in the previous exercise,
and observe that (ax) · y = x · (aT y) for all x, y ∈ A and all a ∈ GLn (k). Let G be a
symmetric subgroup of GLn (k) where “symmetric” means that GT = G. Let
X = Gx : x ∈ A
be the set consisting of all G-orbits on A (with respect
action x 7→ ax).
to the natural
◦
On the other hand, consider the dual group A = ϑx : x ∈ A (as in the previous
exercise). For every x ∈ A define
X
χx =
ϑy , and Y = χx : x ∈ A .
y∈Gx
Prove that (X , Y) is a supercharacter theory for A (note that the natural action of G
◦
on the left of AP
is given by aϑx = ϑa−T x where a−T denotes the inverse transpose of a,
and that χx = ϑ∈Gϑx ϑ for all x ∈ A).
1.4. ([16, Subsection 4.3]) In the notation of the previous exercise, let p be an odd
prime, let A = Fp+ , and let G = hα2 i where α ∈ A is a a primitive root modulo p.
Show that A has three superclasses {0}, G and αG, and three supercharacters χ0 , χ1
and χ2 with supercharacter table given by
χ0
χ1
χ2
{0} G αG
1
1 1
p−1
η0 η1
2
p−1
η1 η2
2
14
CARLOS A. M. ANDRÉ
where
η0 =
X
exp
2πiβ
p
β∈G
and η1 =
X
exp
2πiαβ
p
β∈G
are the usual quadratic Gaussian periods.
1.5. ([16, Subsection 4.4]) In the same notation, let p be an odd prime, let A = Fp ×Fq ,
and let
x 0
×
: x ∈ Fp
G=
0 x−1
where F×
p denotes the multiplicative group of Fp .
(1) Show that the action of G on A produces the superclasses
1 ≤ c < p,
Xc = (x, cx−1 ) : x ∈ F×
p ,
Xp = (0, c) : 1 ≤ c < p − 1 ,
Xp+1 = (c, 0) : 1 ≤ c < p − 1 ,
Xp+2 = {(0, 0)},
and that the supercharacter table is given by
χ1
χ2
..
.
χp−1
χp
χp+1
χp+2
X1
X2
K1
K2
K2
K4
..
..
.
.
Kp−1 K2(p−1)
−1
−1
−1
−1
1
1
where
Kc =
· · · Xp−1
Xp Xp+1
· · · Kp−1
−1
−1
· · · K2(p−1) −1
−1
..
..
..
.
.
.
· · · K(p−1)2 −1
−1
···
−1
p − 1 −1
···
−1
−1 p − 1
···
1
1
1
X
exp
x∈F×
p
Xp+2
p−1
p−1
..
.
p−1
p−1
p−1
1
2πi(x+cx−1 ) p
for all 1 ≤ c < p. [For every a, b ∈ Fp , the sum
X
−1 ) K(a, b) =
exp 2πi(ax+bx
p
x∈F×
p
is the well-known Kloosterman sum. It is easy to see that Kloosterman sums
are always real and that K(a, b) = K(1, ab) = Kab whenever a 6= 0.]
(2) Use the properties of supercharacters to conclude that the matrix
"
#
p
1 χi (Xj ) |Xj |
p
U=
p
|Xi |
1≤i,j≤p+2
is real and symmetric. [This matrix was considered in [20, Eq. (3.13)], and various identities involving Kloosterman sums may be derived from its unitarity.]
1.6. ([21] and [16, Subsection 4.5]) Let n ∈ N, let G = (Z/nZ)× be the group of units
of Z/nZ, and consider the natural action of G on Z/nZ given by multiplication;
notice
that for every a ∈ Z/nZ we have Ga = b ∈ Z/nZ : (a, n) = (b, n) where (x, y)
denotes the greatest common divisor of x and y. Moreover, let τ (n) denote the number
of divisors of n, and let d1 , . . . , dτ (n) be the divisors of n.
SUPERCHARACTERS AND SET PARTITION COMBINATORICS
15
2πira
for all 1 ≤ r ≤
(1) Show that irr(Z/nZ) = {ϑ1 , . . . , ϑn } where ϑr (a) = exp n
n and all a ∈ Z/nZ.
(2) Show that the action of G on Z/nZ produces a supercharacter theory where
the superclasses are X1 , . . . , Xτ (n) where
Xi = a ∈ Z/nZ : (a, n) = n/di
for 1 ≤ i ≤ τ (n), and the supercharacters are χ1 , . . . , χτ (n) where
X
X
χi (a) =
exp 2πika
ϑr (a) =
di
1≤r≤n
(r,i)=n/di
1≤k≤di
(k,di )=1
for 1 ≤ i ≤ τ (n). [This sums are known as the Ramanujan sums and are usually
denoted by cdi (a) for all 1 ≤ i ≤ τ (n) and all a ∈ Z/nZ.]
(3) Using the orthogonality of supercharacters, deduce the following identities where φ X
denotes the Euler totient function:
(a)
φ(k)cdi (n/k)cdj (n/k) = δi,j nφ(di );
k|n
X 1
n
(b)
ck (n/di )ck (n/dj ) = δi,j
;
φ(k)
φ(di )
k|n
X
(c)
φ(k)cd (n/k) = δd,1 n.
k|n
2. Supercharacters and superclasses for algebra groups
2.1. Algebra groups. Let k be a finite field of order q and characteristic p, and let A
be a finite-dimensional nilpotent associative k-algebra; hence, Am = 0 for some m ∈ N,
and in particular A does not have an identity. Let G = 1 + A be the set of formal
objects of the form 1 + a for a ∈ A. Then, G is easily seen to be a group with respect
to the natural multiplication (1 + a)(1 + b) = 1 + a + b + ab. (In fact, G is a subgroup
of the group of units of the k-algebra k · 1 + A in which A is the Jacobson radical.)
A group G constructed in this way is referred to as an algebra group over k (we also
say that G = 1 + A is the algebra group based on A); observe that G is a p-group with
order |G| = |A| = q d where d = dimk (A). For example, if A = un (k) is the algebra of
strictly upper triangular n × n matrices over k, then the corresponding algebra group
G = 1 + A is (isomorphic to) the upper unitriangular group Un (k). Henceforth, we
view G as a subgroup of the group of units of the k-algebra k · 1 + A.
If G = 1 + A is an algebra group over k, then a subgroup H of G is said to be an
algebra subgroup if H = 1 + B for some subalgebra B of A; similarly, a subgroup H of
G is called an ideal subgroup if H = 1 + I for some (two-sided) ideal I of A. It is clear
that an ideal subgroup is always a normal subgroup. For example, for every k ∈ N, the
kth power Ak of A is a two-sided ideal of A, and thus 1 + Ak is an ideal subgroup of G;
recall that by definition Ak is the k-vector subspace of A spanned by all the products
a1 a2 · · · ak of n elements of A. We have a descending chain A ⊇ A2 ⊇ A3 ⊇ . . . of
ideals of A, and hence also a corresponding descending chain of ideal subgroups of G;
notice that, since A is nilpotent, we have Am = 0 for some m ∈ N. We also note
that A2 6= A unless A = 0; in fact, we have the following fact which appears in [27,
Lemma 3.1] (and is an easy consequence of the well-known Nakayama’s Lemma).
Lemma 2.1. Let A be a nilpotent algebra over k. If B ⊆ A is a subalgebra such that
B + A2 = A, then B = A. In particular, A = A2 if and only if A = 0.
16
CARLOS A. M. ANDRÉ
Proof. On the one hand, since Bn ⊆ B, we have An = (B + A2 )n ⊆ B + An+1 , and thus
B + An ⊆ B + An+1 . Therefore, A = B + A2 = B + A3 = · · · , and the result follows
because An = 0 for sufficiently large n.
Several algebra subgroups of G = 1+A appear in connection with the natural actions
of G on A; indeed, left multiplication naturally defines an action of G on the left of
A, whereas right multiplication defines an action of G on the right of A. For every
a ∈ A, we write L(a) to denote the left centraliser of a in G, and R(a) to denote the
right centraliser of a in G; hence,
L(a) = g ∈ G : ga = a
and R(a) = g ∈ G : ag = a .
Both L(a) and R(a) are algebra subgroups of G; in fact, we have L(a) = 1 + L(a) and
Ra = 1 + Ra where
L(a) = b ∈ A : ba = 0
and Ra = b ∈ A : ab = 0
are respectively the left annihilator and the right annihilator of a in A (which are
clearly subalgebras of A). Notice also that both |L(a)| = |L(a)| and |Ra | = |Ra | are
powers of q = |k|, and thus both the left orbit Ga and the right orbit aG have q-power
size; recall that |Ga| = |G|/|L(a)|, and |aG| = |G|/|Ra|.
2.2. Superclasses. Let G = 1 + A be an arbitrary algebra group over k. Since the
left action of G on A obviously commutes with the right action (in the sense that
(ga)h = g(ah) for all g, h ∈ G and all a ∈ A), we can define an action of the group
G × G on the left of A by the rule (g, h) · a = gah−1 for all g, h ∈ G and all a ∈ A.
Thus, the k-algebra A is partitioned into “two-sided” orbits GaG for a ∈ A, and this
determines a partition of group G = 1 + A into subsets Ka = 1 + GaG for a ∈ A;
these are precisely what we define as the superclasses of G. On the other hand, we also
mention the conjugation action of G on A given by g · a = gag −1 for all g ∈ G and all
a ∈ A; we refer to the corresponding orbits as the conjugation orbits of G on A. We
note that, since g(1 + a)g −1 = 1 + gag −1 for all g ∈ G and all a ∈ A, the conjugation
orbits of G on A are in one-to-one correspondence with the conjugacy classes of G;
moreover, it is also clear that every two-sided orbit GaG is a union of conjugation
orbits, and hence every superclass of G is a union of conjugacy classes.
Lemma 2.2. Let G = 1 + A be an algebra group over k. If a ∈ A, then
|GaG| =
|Ga||aG|
.
|Ga ∩ aG|
In particular, the superclasses of G all have q-power size where q = |k|.
Proof. For any a ∈ A, we define the map φ : Ga × aG → GaG by φ(ga, ah) = gah for
all g, h ∈ G. It is straightforward to check that φ is well-defined, and that all its fibres
have the same cardinality |φ−1 (a)| = |Ga ∩ aG|. Since φ is obviously surjective, we
conclude that |GaG||Ga ∩ aG| = |Ga||aG|. On the other hand, we have Ga = a + Aa,
aG = a + aA and Ga ∩ aG = a + (Aa ∩ aA). Since Aa, aA and Aa + aA are k-vector
subspaces of A, it follows that |Ga|, |aG| and |Ga ∩ aG| are powers of q. The result
now follows.
SUPERCHARACTERS AND SET PARTITION COMBINATORICS
17
2.3. G-actions on the dual group A◦ . Our next goal is to define the supercharacters
of an algebra group G = 1 + A. For this, we consider the dual group A◦ of the additive
group A+ of A. For each of the actions of G on A which we introduced above, there
is a corresponding action of G on A◦ . Given ϑ ∈ A◦ and g ∈ G, we define gϑ, ϑg ∈ A◦
by the formulas
(gϑ)(a) = ϑ(g −1 a) and (ϑg)(a) = ϑ(g −1 a)
for all a ∈ A. It is routine to check that these actions commute, and thus we have
left orbits Gϑ, right orbits ϑG, and also “two-sided” orbits GϑG for any ϑ ∈ A◦ . In
addition, we also mention that there is the conjugation action of G on A◦ defined by
(g, ϑ) → g −1 ϑg.
Lemma 2.3. Let G = 1 + A be an algebra group over k. Then, the numbers of left
orbits of G on A and on A◦ are equal. The same is true about right orbits, two-sided
orbits and conjugation orbits.
Proof. Since A is abelian (as an additive group), the conjugacy classes are singleton
sets. Since (gϑ)(ga) = ϑ(a) for all g ∈ G, all ϑ ∈ A◦ and a ∈ A◦ , Brauer’s theorem
([26, Theorem 6.32]) asserts that for every g ∈ G the number of elements fixed by
g on A equals the number of elements fixed by g on A◦ . The result follows by the
orbit-counting formula (1f).
As in the proof of Lemma 2.2, for each ϑ ∈ A◦ we may define a surjective map
ψ : Gϑ × ϑG → GϑG by ψ(gϑ, ϑh) = gϑh for all g, h ∈ G; this map is in fact welldefined, and all its fibres have the same cardinality |φ−1 (ϑ)| = |Gϑ ∩ ϑG|. This proves
the following.
Lemma 2.4. Let G = 1 + A be an algebra group over k. If ϑ ∈ A, then
|GϑG| =
|Gϑ||ϑG|
.
|Gϑ ∩ ϑG|
For every ϑ ∈ A◦ , we define
L(ϑ) = g ∈ G : gϑ = ϑ
and R(ϑ) = g ∈ G : ϑg = ϑ ;
hence L(ϑ) is the left centraliser of ϑ in G, and R(ϑ) is the right centraliser of ϑ in G.
On the other hand, we define
L(ϑ) = a ∈ A : aA ⊆ ker(ϑ)
and R(ϑ) = a ∈ A : Aa ⊆ ker(ϑ) ;
it is routine to check that both L(ϑ) and R(ϑ) are subalgebras of A. Moreover, we have
L(ϑ) = 1 + L(ϑ) and R(ϑ) = 1 + R(ϑ), hence L(ϑ) and R(ϑ) are algebra subgroups of
G. It follows that both |L(ϑ)| = |L(ϑ)| and |R(ϑ)| = |R(ϑ)| are powers of q = |k|, and
thus both the left orbit Gϑ and the right orbit ϑG have q-power size.
2.4. Supercharacters. For notational simplicity, we introduce the (bijective) map
ν : G → A given by ν(g) = g − 1 for all g ∈ G, and observe that
ν(gh) = gh − 1 = gh − g + g − 1 = gν(h) + ν(g)
for all g, h ∈ G. We define a new left action of G on A by setting
(2a)
g · a = ga + ν(g)
for all g ∈ G and all a ∈ A; we note that
g · ν(h) = gν(h) + ν(g) = ν(gh)
18
CARLOS A. M. ANDRÉ
for all g, h ∈ G. This action can be extended by linearity to the group algebra CA of A,
so that CA becomes a CG-module; moreover, it is clear that the map ν : G → A extends
linearly to an isomorphism of CG-modules ν : CG → CA where CG is considered as
the regular CG-module. In particular, we obtain the following.
Lemma 2.5. Let G = 1 + A be an algebra group over k. Then, with respect to the
G-action defined by (2a), the CG-module CA affords the regular character ρG of G.
Besides its natural C-basis, the group-algebra CA also has a C-basis εϑ : ϑ ∈ A◦
where εϑ ∈ A is the (central) primitive idempotent 5
1 X
ϑ(a) a
(2b)
εϑ =
|A| a∈A
◦
corresponding
to the linear character ϑ ∈ A ; it is an easy exercise to show that
◦
εϑ : ϑ ∈ A is indeed a set of mutually orthogonal primitive idempotents (this is
essentially the content of the first orthogonality relations for the irreducible characters
of A). For any ϑ ∈ A◦ , we evaluate
1 X
1 X
g · εϑ =
ϑ(a) (g · a) =
ϑ(g −1 · a) a
|A| a∈A
|A| a∈A
!
X
X
1
1
ϑ(g −1 a + ν(g −1 )) a = ϑ(ν(g −1 ))
ϑ(g −1 a) a
=
|A| a∈A
|A| a∈A
!
X
1
= ϑ(ν(g −1 ))
(gϑ)(a) a = ϑ(ν(g −1 )) εgϑ
|A| a∈A
for all g ∈ G, and thus the left G-orbit Gϑ ⊆ A◦ spans the CG-submodule CG · εϑ of
CA generated by εϑ . We denote by χϑ the character of G afforded by the CG-module
CG · εϑ , and we refer
to these characters as the supercharacters of G; we note that,
since ετ : τ ∈ Gϑ is a C-basis of CG · εϑ , we have
χϑ (1) = dimC (CG · εϑ ) = |Gϑ|.
Henceforth, we simplify the notation and write Lϑ to denote the CG-module CG · εϑ .
Proposition 2.6. Let G = 1 + A be an algebra group over k, let ϑ ∈ A◦ , and let L(ϑ)
denote the left centraliser of ϑ is G. Then, the mapping g 7→ ϑ(g − 1) defines a linear
character ϑ̂ : L(ϑ) → C and we have χϑ = IndG
L(ϑ) (ϑ̂).
Proof. The one-dimensional vector subspace Cεϑ of Lϑ is easily shown to be a CL(ϑ)submodule. Since g · εϑ = ϑ(ν(g −1 ))εgϑ = ϑ(ν(g −1 ))εϑ for all g ∈ L(ϑ), we see that
Cεϑ affords the linear character of L(ϑ) defined by the mapping g 7→ ϑ(ν(g −1 )), and
thus we must prove that ϑ(ν(g −1 )) = ϑ(ν(g)) for all g ∈ L(ϑ). To see this, it is enough
to observe that ν(g −1 ) = g −1 − 1 = −g −1 (g − 1) = −g −1 ν(g), hence
ϑ(ν(g −1 )) = ϑ(−g −1 ν(g)) = ϑ(g −1 ν(g)) = (gϑ)(ν(g)) = ϑ(ν(g))
for all g ∈ L(ϑ).
L
Finally, the CG-module Lϑ decomposes as the direct sum Lϑ = τ ∈Gϑ Cετ where the
summands are transitively permuted by G; without loss of generality, we may choose
5For
any C-algebra, an element e is an idempotent if e2 = e; it is primitive if it is not the sum of
two idempotents; and it is central if it commutes with every element of the algebra. Two idempotents
e and e′ are said to be orthogonal if ee′ = 0.
SUPERCHARACTERS AND SET PARTITION COMBINATORICS
19
one of this direct summands to be Cεϑ . Since L(ϑ) is clearly the stabiliser StabG (Cεϑ )
of Cεϑ in G, we conclude that Lϑ = IndG
L(ϑ) (Cεϑ ), and the result follows.
Our next result is almost obvious and establishes axiom (3) of the definition of a
supercharacter theory.
Proposition 2.7. Let G = 1+A be an algebra group over k, and let ϑ1 , . . . , ϑn ∈ A◦ be
a complete set of representatives for the left orbits of G on A◦ . Then, the CG-module
CA decomposes as the direct sum CA = Lϑ1 ⊕· · ·⊕Lϑn , and hence the regular character
ρG of G decomposes as the sum ρG = χϑ1 + · · · + χϑn . In particular, every irreducible
character χ ∈ irr(G) is a constituent of at least one supercharacter χϑ for ϑ ∈ A◦.
Proof. Since A◦ decomposes as the disjoint union of the orbits Gϑ1 , . . . , Gϑn and since
A◦ is a C-basis of CA, the previous lemma implies that CA has the required direct
sum decomposition. The last assertion follows because every irreducible character of
G is a constituent of the regular character.
2.5. Orthogonality of supercharacters. In view of the previous proposition, in
order to show that the given data form a supercharacter theory for the algebra group
G = 1 + A, it remains to prove that supercharacters are orthogonal and constant
on superclasses, and that its number equals the number of superclasses (which by
Lemma 2.3 equals the number of two-sided G-orbits on A◦ ). Firstly, we prove that
distinct supercharacters are in fact orthogonal.
To see this, we recall that for every
◦
ϑ, τ ∈ A the Frobenius scalar product χϑ , χτ equals the dimension of the vector
space HomCG (Lϑ , Lτ ).
We observe that the group algebra CA also has the structure of a right CG-module
with respect to the right G-action defined by
a · g = ag + ν(g)
for all a ∈ A and all g ∈ G. Since ν(gh) = ν(g)h + ν(h), we see that ν(h) · g = ν(hg)
for all g, h ∈ G, and thus ν : CG → CA is also an isomorphism of right CG-modules
(where CG is now considered as the regular right CG-module). Similarly to the above,
for every ϑ ∈A◦ , we may consider the right CG-submodule Rϑ = εϑ · CG of CA which
has C-basis ετ : τ ∈ ϑG . (As a right CG-module, Rϑ affords a character χ′ϑ of G
which has degree χ′ϑ (1) = |ϑG| = |Gϑ| = χϑ (1); in fact, it can be proved that χ′ϑ equals
the supercharacter χϑ of G.)
Lemma 2.8. Let G = 1 + A be an algebra group over k and let
ϑ ∈ A◦ . Then, there
exists a C-isomorphism HomCG (Lϑ , CA) ∼
= Rϑ . In particular, χϑ , ρG = |ϑG|, and
thus |Gϑ| = |ϑG|.
Proof. By Maschke’s theorem, there exists a CG-module L′ϑ of CA such that CA =
Lϑ ⊕ L′ϑ , and so every φ ∈ HomCG (Lϑ , CA) can be naturally extended to a CGe In virtue of the isomorphism
endomorphism φe ∈ EndCG (CA) satisfying L′ϑ ⊆ ker(φ).
∼
CA =CG CG, there is a C-linear isomorphism EndCG (CA) ∼
=C EndCG (CG), and this
e
implies that there exists a unique z ∈ CG such that φ(ς) = ς · z for all ς ∈ CA. In
e ϑ ) = εϑ · z ∈ Rϑ , and therefore the mapping φ 7→ φ(εϑ ) defines
particular, φ(εϑ ) = φ(ε
a C-linear map HomCG (Lϑ , CG) → Rϑ which is at once seen to be surjective. It is
injective because if φ(εϑ ) = 0, then φ satisfies φ(g · εϑ ) = gφ(εϑ) = 0 for all g ∈ G.
This proves that HomCG (Lϑ , CG) ∼
= Rϑ .
20
CARLOS A. M. ANDRÉ
For the last assertion, we recall that CA affords the regular character of G, and
hence
χϑ , ρG = dimC HomCG (Lϑ , CA) = dimC Rϑ = |ϑG|.
Finally, we deduce that
1 X
ρG (g)χϑ (g) = χϑ (1) = |Gϑ|,
ρG , χϑ =
|G| g∈G
and this completes the proof.
Corollary 2.9. Let G = 1 + A be an algebra group over k and let ϑ, ϑ′ ∈ A◦ . Then,
there is a C-linear isomorphism
HomCG (Lϑ , Lϑ′ ) ∼
= Rϑ ∩ Lϑ′ .
= Gϑ′ G, and hence χϑ , χϑ′ 6=
In particular, HomCG (Lϑ , Lϑ′ ) 6= {0} if and only if GϑG
0 if and only if GϑG = Gϑ′ G. Moreover, we have χϑ , χϑ = |Gϑ ∩ ϑG|.
Proof. By the proof of the previous lemma, we see that φ ∈ HomCG (Lϑ , Lϑ′ ) if and
only if φ(εϑ ) ∈ Rϑ ∩ Lϑ′ , and thus the mapping φ 7→ φ(εϑ ) restricts to a C-linear
isomorphism HomCG (Lϑ , Lϑ′ ) ∼
= Rϑ ∩ Lϑ′ . In particular, we deduce that
χϑ , χϑ′ = dimC HomCG (Lϑ , Lϑ′ ) = dimC (Rϑ ∩ Lϑ′ ).
Since ετ : τ ∈ Gϑ ∩ ϑ′ G is a C-basis of Lϑ ∩ Rϑ′ , we conclude that Lϑ ∩ Rϑ′ 6= 0
if and only if Gϑ ∩ ϑ′ G 6= ∅. The result follows because Gϑ ∩ ϑ′ G 6= ∅ if and only if
ϑ′ ∈ GϑG.
2.6. Supercharacter values. At this point, we know the supercharacters of G = 1+A
are orthogonal and that their number is equal to the number of two-sided G-orbits on
A◦ . By Lemma 2.3, this number is equal to the number of two-sided G-orbits on A
which in turn is equal to the number of superclasses of G. Therefore, to establish
that we have a genuine supercharacter theory it remains to show that supercharacters
are constant on superclasses. To prove this we need a convenient way to compute the
values of a supercharacter.
Keeping the notation as above, let ϑ∈ A◦ be arbitrary, and consider the vector
subspace Dϑ = Lϑ ∩ Rϑ of CA; then, ετ : τ ∈ Gϑ ∩ ϑG is a C-basis of Dϑ , and
hence dimC D
ϑ = |Gϑ ∩ ϑG|. We define the subgroup Sϑ of G to be the stabiliser
StabG (Dϑ ) = g ∈ G : g · Dϑ ⊆ Dϑ of Dϑ in G; notice that g · Dϑ = Dgϑ for all g ∈ G.
If CSϑ denotes the group algebra of Sϑ , then Dϑ is a CSϑ -module, and thus it affords
a character ςϑ of Sϑ . We have the following.
Lemma 2.10. Let G = 1 + A be an algebra group over k and let ϑ ∈ A◦ . Then, the
supercharacter χϑ of G is induced by the character ςϑ of Sϑ afforded by the CSϑ -module
Dϑ .
Proof. If T ⊆ G is a complete set of representatives
of the left cosets of Sϑ in G,
S
then G decomposes as the disjoint unionS
G = t∈T tSϑ , and hence the left G-orbit Gϑ
decomposes as the disjoint union Gϑ = t∈T t(Gϑ ∩ ϑG). Therefore, the CG-module
Lϑ decomposes as the direct sum
M
M
Lϑ =
t · Dϑ =
Dtϑ
t∈T
t∈T
of vector subspaces where the summands are transitively permuted by G. Since Sϑ =
StabG (Dϑ ), we conclude that Lϑ = IndG
Sϑ (Dϑ ), and the result follows.
SUPERCHARACTERS AND SET PARTITION COMBINATORICS
21
Lemma 2.11. Let G = 1 + A be an algebra group over k and let ϑ ∈ A◦ . Then,
|Gϑ| X
|Gϑ| X
τ (ν(g)) =
τ (g − 1)
ςϑ◦ (g) =
|GϑG| τ ∈ϑG
|GϑG| τ ∈ϑG
for all g ∈ G.
Proof. For every g ∈ G, we evaluate
X
|ϑG| X
|ϑG| X
τ (ν(g)) =
(ϑh−1 )(ν(g)) =
ϑ ν(g)h
|G| h∈G
|G| h∈G
τ ∈ϑG
|ϑG| X
|ϑG| X
=
ϑ ν(gh) − ν(h) =
ϑ ν(gh) ϑ ν(h)
|G| h∈G
|G| h∈G
|ϑG| X
=
ϑ gν(h) + ν(g) ϑ ν(h)
|G| h∈G
|ϑG| ϑ(ν(g)) X −1
(g ϑ)(ν(h))ϑ(ν(h)).
=
|G|
h∈G
Since ϑ, g −1ϑ ∈ A◦ and since ν : G → A is bijective, we have
(
X
1
1, if g −1 ϑ = ϑ,
(g −1 ϑ)(ν(h))ϑ(ν(h)) = g −1ϑ, ϑ A =
|G| h∈G
0, if g −1 ϑ 6= ϑ,
and thus
(
|ϑG| ϑ(ν(g)), if gϑ = ϑ,
τ (ν(g)) =
0,
if gϑ 6= ϑ.
τ ∈ϑG
X
If g ∈
/ Sϑ , then clearly gϑ 6= ϑ, and hence the required equality holds in this case.
On the other hand, we assume that g ∈ Sϑ , and evaluate ςϑ (g) = Tr(g, Dϑ ). For any
τ ∈ Gϑ ∩ ϑG, we have g · ετ = τ (ν(g −1 ) εgτ , and so
X
τ (ν(g −1 ).
ςϑ (g) =
τ ∈Gϑ∩ϑG
gτ =g
Since τ ∈ ϑG, it is easy to see that gτ = τ if and only if gϑ = ϑ, and hence ςϑ (g) 6= 0
if and only if gϑ = ϑ. Therefore, it remains to prove that
ςϑ (g) =
|Gϑ| |ϑG|
ϑ(ν(g)) = |Gϑ ∩ ϑG| ϑ(ν(g))
|GϑG|
whenever g ∈ Sϑ is such that gϑ = ϑ. Indeed, if this is the case, we may repeat the
calculations above to check that
(ϑh) ν(g −1 ) = ϑ ν(g −1 )) (gϑ) ν(h−1 ) ϑ ν(h−1 )
= ϑ ν(g −1 )) ϑ ν(h−1 ) ϑ ν(h−1 ) = ϑ ν(g −1 ))
for all h ∈ G, and this implies that
X
ςϑ (g) =
τ (ν(g −1 ) = |Gϑ ∩ ϑG| ϑ ν(g −1 ) .
τ ∈Gϑ∩ϑG
Finally, since ν(1) = 0, we have
ϑ(ν(g)) = (gϑ)(ν(g)) = ϑ(g −1 ν(g)) = ϑ ν(g −1 g) − ν(g −1 ) = ϑ(ν(g −1 )),
22
CARLOS A. M. ANDRÉ
and so
This completes the proof.
ςϑ (g) = |Gϑ ∩ ϑG| ϑ ν(g) .
We are now able to prove the main formula for supercharacter values.
Theorem 2.12. Let G = 1 + A be an algebra group over k, and let ϑ ∈ A◦ . Then,
|Gϑ| X
τ (g − 1)
(2c)
χϑ (g) =
|GϑG| τ ∈GϑG
for all g ∈ G.
Proof. Let g ∈ G be arbitrary, and let T ⊆ G be is a complete set of representatives of
the left cosets of Sϑ in G. Then, by Lemmas 2.10 and 2.11, we obtain
X
|Gϑ| X X
χϑ (g) =
ςϑ◦ (t−1 gt) =
τ (ν(t−1 gt)).
|GϑG|
t∈G
t∈G τ ∈ϑG
Since ν(t−1 gt) = t−1 ν(g)t and since the mapping τ 7→ tτ t−1 defines a bijection between
ϑG and tϑGt−1 = tϑG, we deduce that
X
X
X
X
τ (ν(g))
(tτ t−1 )(ν(g)) =
τ (t−1 ν(g)t) =
τ (ν(t−1 gt)) =
τ ∈ϑG
τ ∈ϑG
τ ∈ϑG
τ ∈tϑG
for all t ∈ T . Since Sϑ ϑ ⊆ ϑG, it is easy
S to see that the two-sided G-orbit GϑG
decomposes as the disjoint union GϑG = t∈T tϑG, and so we conclude that
χϑ (g) =
as required.
|Gϑ| X X
|Gϑ| X
τ (ν(g)) =
τ (ν(g))
|GϑG| t∈G τ ∈tϑG
|GϑG| τ ∈GϑG
As a consequence, we see that supercharacters are constant on superclasses. In fact,
our next result provides a formula for the supercharacter values on each superclasses;
we recall that by definition the superclass containing an element g ∈ G is the set
K = 1 + Gν(g)G = 1 + G(g − 1)G.
Corollary 2.13. Let G = 1 + A be an algebra group over k, let ϑ ∈ A◦ and let g ∈ G.
Then,
χϑ (1) X
χϑ (g) =
ϑ(h − 1)
|K| h∈K
where K ⊆ G is the superclass which contains g. In particular, the supercharacters of
G are constant on the superclasses.
Proof. Since Gπ(g)G and GϑG are orbits for the action of G × G on A and on A◦
respectively, we deduce that
X
X
X
|G|2
|G|2
ϑ(xν(g)y) =
(x−1 ϑy −1 )(ν(g))
ϑ(ν(h)) =
|Gν(g)G| x,y∈G
|Gν(g)G| x,y∈G
h∈K
=
|GϑG| X
|G|2
|GϑG| X
ς(ν(g)).
ς(ν(g))
=
|Gπ(g)G| |G|2 ς∈GϑG
|Gν(g)G| ς∈GϑG
SUPERCHARACTERS AND SET PARTITION COMBINATORICS
By the previous theorem, we conclude that
|Gϑ| X
|Gϑ| X
ς(ν(g)) =
ϑ(ν(h)),
ξϑ (g) =
|GϑG| ς∈G
|Gν(g)G| h∈K
as required.
23
Another consequence is that the definition of the supercharacters of G is the same
if we consider the right G-orbits in A◦ instead of the left G-orbits, hence replacing the
left CG-submodules Lϑ by the right CG-submodules Rϑ = εϑ · CG of CA. In fact, the
following dual version of Proposition 2.6 holds.
Proposition 2.14. Let G = 1 + A be an algebra group over k, let ϑ ∈ A◦ , and let
R(ϑ) denote the right centraliser of ϑ is G. Then, the mapping g 7→ ϑ(g − 1) defines a
linear character ϑ̂ : R(ϑ) → C and the induced character IndG
R(ϑ) (ϑ̂) is afforded by the
right CG-submodule Rϑ = εϑ · CG of CA.
Repeating the argument above, we then obtain the formula
|ϑG| X
IndG
(
ϑ̂)(g)
=
τ (g − 1)
R(ϑ)
|GϑG| τ ∈GϑG
which we see to be precisely (2c) because |ϑG| = |Gϑ| (by Lemma 2.8; see also Exercise 2.3).
Proposition 2.15. Let G = 1 + A be an algebra group over k, and let ϑ ∈ A◦ . Then,
the right CG-submodule Rϑ = εϑ · CG of CA affords the supercharacter χϑ of G, and
thus χϑ = IndG
R(ϑ) (ϑ̂) where R(ϑ) and ϑ̂ : R(ϑ) → C are as in the previous lemma.
2.7. Restriction, products and superinduction of supercharacters. As above,
we let G = 1 + A be an algebra group over k. If B ⊆ A is a subalgebra, then the
algebra subgroup H = 1 + B of G is itself an algebra group over k. Our first goal in
this section is to show that the restriction to H of a supercharacter of G is a linear
combination of supercharacters of H with nonnegative integer coefficients. We first
observe that the vector space scf(G) consisting of all complex-valued functions on G
that are constant on superclasses has an obvious distinguished C-basis, the superclass
characteristic functions; if K is a superclass, then the characteristic function κ : G → C
is defined by
(
1, if g ∈ K,
κ(g) =
0, if g ∈
/ K.
Another distinguished basis of scf(G) consists of all supercharacters; we recall that we
are writing sch(G) to denote the set of supercharacters of G. In particular, since the
restriction χH = ResG
H (ξ) of any supercharacter χ ∈ sch(G) is clearly constant on the
superclasses of H, it is a C-linear combination of supercharacters of H, that is,
X
ResG
(χ)
=
cζ ζ
H
ζ∈sch(H)
where cζ ∈ C for ζ ∈ sch(H). Our claim is that the coefficients cζ are nonnegative
integers; our proof will be by induction on |G|, and the induction step depends of the
following consequence of Lemma 2.1.
Lemma 2.16. If B is a maximal subalgebra of the nilpotent k-algebra A, then B contains A2 , and B is an ideal of A with codimension one.
24
CARLOS A. M. ANDRÉ
Proof. Since B A, Lemma 2.1 yields B ⊆ B + A2 A. Since A2 is an ideal, B + A2
is a subalgebra of A, and thus B = B + A2 by the maximality of B. In particular, we
have A2 ⊆ B, and the result follows because every subspace of A containing A2 is an
ideal of A.
In the following, we fix a maximal subalgebra B of A, and consider the algebra
subgroup H = 1 + B of G = 1 + A. Let ϑ ∈ A◦ be arbitrary, and recall that the
supercharacter
χϑ is afforded by the CG-submodule Lϑ = CG · εϑ of CA. Recall also
that ετ : τ ∈ Gϑ is a C-basis of L
g · ετ = τ (g −1 − 1) εgτ for all g ∈ G and
ϑ , and that all τ ∈ Gϑ. In particular, the set ετ : τ ∈ Hϑ spans the CH-submodule CH · εϑ of
Lϑ which we denote by L′ϑ . Let ζϑ denote the character of H afforded by L′ϑ , and note
that ζϑ is a constituent of the restriction ResG
H (χϑ ) of χϑ to H. Let g ∈ G be arbitrary.
Then, since H is a normal subgroup
of
G
(because B is an ideal of A), the vector
′
′
′
subspace g · Lϑ = g · v : v ∈ Lϑ of Lϑ is clearly a CH-submodule, and it affords the
conjugate character (ζϑ )g of H which is defined by (ζϑ )g (h) = ζϑ (g −1 hg)
for all h ∈ H.
In fact, it is routine to check that g · L′ϑ = CH · εgϑ = L′gϑ ; hence, ετ : τ ∈ H(gϑ)
is a C-basis of g · L′ϑ , and (ζϑ )g = ζgϑ . Since Gϑ decomposes as the disjoint union
Gϑ = H(g1ϑ) ∪ · · · ∪ H(gr ϑ) for some g1 , . . . , gr ∈ G, we obtain a decomposition
Lϑ = L′g1 ϑ ⊕ · · · ⊕ L′gr ϑ
of Lϑ as a direct sum of CH-submodules, and thus
ResG
H (χϑ ) = ζg1 ϑ + · · · + ζgr ϑ .
We now prove the following result.
Proposition 2.17. Let G = 1 + A be an algebra group over k, and let H = 1 + B be a
maximal algebra subgroup of G. Let ϑ ∈ A◦ , and let L(ϑ) be the left centraliser of ϑ in
G. Then, either L(ϑ) ⊆ H or HL(ϑ) = G. Moreover, if L′ϑ = CH · εϑ and ζϑ denotes
the character of H afforded by L′ϑ , then the following hold.
G
(1) If HL(ϑ) = G, then Gϑ = Hϑ; in particular, Lϑ = L′ϑ and Res
S H (χϑ ) = ζϑ .
(2) If L(ϑ) ⊆ H, then Gϑ decomposes as the disjoint union Gϑ = t∈T H(tϑ) where
T is aL
complete set of representatives
P of the left cosets of H in G; in particular,
G
′
Lϑ = t∈T Ltϑ and ResH (χϑ ) = t∈T ζtϑ .
Proof. We know that L(ϑ) = 1 + L(ϑ) where L(ϑ) is a subalgebra of A. Since A2 ⊆ B,
the vector space B+L(ϑ) is a subalgebra of A, and thus either L(ϑ) ⊆ B or B+L(ϑ) = A
(by the maximality of B). It follows that either L(ϑ) ⊆ H or HL(ϑ) = G; notice that
HL(ϑ) = 1+(B+L(ϑ)). If HL(ϑ) = G, then Gϑ = HL(ϑ)ϑ = Hϑ, hence Lϑ = L′ϑ and
ResG
H (χϑ ) = ζϑ . On the other hand, assume that L(ϑ) ⊆ H. Then, L(ϑ) = L(ϑ) ∩ H
is the left centraliser
of ϑ in H, and thus |Gϑ| = |G|/|L(ϑ)| = q|H|/|L(ϑ)| = q|Hϑ|.
S
Since Gϑ = t∈T H(tϑ) and |T | = |G : H| = q, it follows that this union is disjoint,
L
and this clearly implies Lϑ = t∈T L′tϑ (because ετ : τ ∈ H(tϑ) is a C-basis of L′tϑ ).
The result follows.
Keeping the notation as above, we now relate the CH-modules L′ϑ = CH · εϑ and
Lϑ0 = CH · εϑ0 where ϑ0 denotes the restriction of ϑ to B. Since B has codimension
one, there is a one-dimensional vector subspace V of A such that A = B ⊕V, and hence
there is a natural k-linear map π : A → B given by π(b + v) = b for all b ∈ B and all
v ∈ V. Therefore, by linear extension we obtain a C-linear map π : CA → CB. Since
B is an ideal of A, we have gb ∈ B for all g ∈ G and all b ∈ B, and hence the group
algebra CB has an obvious structure of CG-module; moreover, since π(ga) = gπ(a)
SUPERCHARACTERS AND SET PARTITION COMBINATORICS
25
for all g ∈ G and all a ∈ A, the map π : CA → CB is in fact a homomorphism of
CG-modules. By the definition of εϑ , it is easily seen that π(εϑ ) = q −1 εϑ0 , and thus
π(L′ϑ ) = Lϑ0 ; in fact, we have Lgϑ0 = π(L′gϑ ) = π(g · L′ϑ ) = g · Lϑ0 for all g ∈ G.
On the other hand, let ρ : A◦ → B◦ denote the natural projection which sends every
◦
character τ ∈ A◦ to its restriction to B; hence, ρ(τ ) = ResA
B (τ ) for all τ ∈ A . It
is easily seen that ρ is a left G-invariant surjective homomorphism of groups whose
kernel is the orthogonal group B⊥ of B. In particular, since ρ(Hϑ) = Hϑ0 , we conclude
that |Hϑ| ≤ q|Hϑ0 |, and thus dimC L′ϑ = |Hϑ| ≤ q|Hϑ0 | = q dimC Lϑ0 . Furthermore,
if CH (ϑ) and CH (ϑ0 ) denote the left H-centralisers of ϑ and ϑ0 respectively, then
|H| = |CH (ϑ)| |Hϑ| and |H| = |CH (ϑ0 )| |Hϑ0|, and so |CH (ϑ)| ≤ q |CH (ϑ0 )|. Since
CH (ϑ) ⊆ CH (ϑ0 ) and both CH (ϑ) and CH (ϑ0 ) are algebra subgroups of H (exercise),
we conclude that either CH (ϑ) = CH (ϑ0 ) or CH (ϑ) is a subgroup of CH (ϑ0 ) with index
|CH (ϑ0 ) : CH (ϑ)| = q. It follows that either |Hϑ| = |Hϑ0 | or |Hϑ| = q|Hϑ0 |, and this
implies that either dimC L′ϑ = dimC Lϑ0 or dimC L′ϑ = q dimC Lϑ0 .
We are now able to prove the following.
Lemma 2.18. Let G = 1 + A be an algebra group over k, and let H = 1 + B be a
maximal algebra subgroup of G. Let ϑ ∈ A◦ , let ϑ0 ∈ B◦ denote the restriction of ϑ to B,
and suppose that |Hϑ| = |Hϑ0 |. Then, the restriction ResG
H (χϑ ) is a linear combination
with positive integers coefficients of supercharacters of H, and the following hold.
(1) If g ∈ G, then the supercharacter χgϑ0 of H is a constituent of ResG
H (χϑ );
G
(2) If χ ∈ sch(H) is a constituent of ResH (χϑ ), then χ = χgϑ0 for some g ∈ G.
Proof. Since dimC L′ϑ = |Hϑ| = |Hϑ0 | = dimC Lϑ0 , the projection π : CA → CB
restricts to an isomorphism of CH-modules π : L′ϑ → Lϑ0 , and so ζϑ = χϑ0 . The result
follows from Proposition 2.17 because Lgϑ0 = π(L′gϑ ) and |Hgϑ| = |gHϑ| = |Hϑ| =
|Hϑ0 | = |gHϑ0| = |Hgϑ0 | for all g ∈ G.
Replacing “left” by “right” in the previous argument, we obtain the “dual” result.
Lemma 2.19. Let G = 1 + A be an algebra group over k, and let H = 1 + B be a
maximal algebra subgroup of G. Let ϑ ∈ A◦ , let ϑ0 ∈ B◦ denote the restriction of ϑ
to B, and suppose that |ϑH| = |ϑ0 H|. Then, the restriction ResG
H (χϑ ) decomposes a
linear combination with positive integers coefficients of supercharacters of H, and the
following hold.
(1) If g ∈ G, then the supercharacter χϑ0 g of H is a constituent of ResG
H (χϑ );
G
(2) If χ ∈ sch(H) is a constituent of ResH (χϑ ), then χ = χϑ0 g for some g ∈ G.
It remains to consider the case where |Hϑ| = q |Hϑ0 | and |ϑH| = q |ϑ0 H|; we recall
that |Hϑ0 | = |ϑ0 H| (by Lemma 2.8), and hence |Hϑ| = |ϑH|. In this situation, we
have the following.
Lemma 2.20. Let G = 1 + A be an algebra group over k, and let H = 1 + B be a
maximal algebra subgroup of G. Let ϑ ∈ A◦ , let ϑ0 ∈ B◦ denote the restriction of ϑ to B,
and suppose that |ϑH| = q |ϑ0 H| and |ϑH| = q |ϑ0 H|. Then, the restriction ResG
H (χϑ )
decomposes a linear combination with positive integers coefficients of supercharacters
of H, and the following hold.
(1) If g ∈ G, then the supercharacter χgϑ0 of H is a constituent of ResG
H (χϑ );
G
(2) If χ ∈ sch(H) is a constituent of ResH (χϑ ), then χ = χgϑ0 for some g ∈ G.
Proof. Let L = CH (ϑ) and L0 = CH (ϑ0 ) be the left H-centralisers of ϑ and ϑ0 respectively, and note that L is a subgroup of L0 with index q (because |Hϑ| = q |Hϑ0 |).
26
CARLOS A. M. ANDRÉ
Moreover,
we
have
L
=
1
+
L
for
L
=
b
∈
B
:
bA
⊆
ker(ϑ)
, whereas L0 = 1 + L0 for
L0 = b ∈ B : bB ⊆ ker(ϑ0 ) ; notice that both L and L0 are subalgebras of B.
We recall from Proposition 2.6 that χϑ0 = IndH
C0 (ϑ̂0 ) where ϑ̂0 is the linear character
of L0 defined by ϑ̂0 (h) = ϑ0 (h − 1) for all h ∈ L0 . On the other hand, the onedimensional vector space Cεϑ is clearly a CL-submodule of L′ϑ which affords the linear
character
ϑ̂ = ResLL0 (ϑ̂0 ) of L. Since the CH-module L′ϑ decomposes as the direct sum
L
L′ϑ = τ ∈Hϑ Cετ where the summands are transitively permuted by H, and since L is
easily seen to be the stabiliser StabH (Cεϑ ), it follows that ζϑ = IndH
L (ϑ̂); recall that the
′
CH-module Lϑ affords the character ζϑ of H. Since ϑ̂0 is a constituent of the induced
character IndLL0 (ϑ̂) (by Frobenius reciprocity), we conclude that the supercharacter
H
L0
H
χϑ0 = IndH
L0 (ϑ̂0 ) of H is a constituent of IndL0 IndL (ϑ̂) = IndL (ϑ̂) = ζϑ , and hence
H
G
it is a constituent of the supercharacter χϑ = IndG
L (ϑ̂) = IndH IndL (ϑ̂) of G.
In order to obtain the complete decomposition of ζϑ as a sum of supercharacters of
H, we now consider the orthogonal subgroup L⊥ of L in B◦ . Let τ ∈ L⊥ be arbitrary.
d
Then, (ϑ0 τ )(a) = ϑ0 (a) for all a ∈ L, and thus ϑ
0 τ is a linear character of L0 satisfying
L0 d
d
ResL (ϑ0 τ ) = ϑ̂. By Frobenius reciprocity, it follows that ϑ
0 τ is a constituent of the
L0
H
induced character IndL (ϑ̂), and thus IndL0 (ϑ0 τ ) is a constituent of IndH
L (ϑ̂) = ζϑ . In
fact, it is not hard to prove that there are q = |L0 : L| linear characters τ1 , . . . , τq ∈ L⊥
L0
d
such that ϑd
0 τ1 , . . . , ϑ0 τq are distinct constituents of the induced character IndL (ϑ̂).
d
Since IndLL0 (ϑ̂) has degree |L0 : L| = q, we conclude that IndLL0 (ϑ̂) = ϑd
0 τ1 + · · · + ϑ0 τq ,
and so
H d
d
ζϑ = IndH
L0 (ϑ0 τ1 ) + · · · + IndL0 (ϑ0 τq ).
Therefore, to achieve our goal it is now enough to prove that for every τ ∈ L⊥ the
H c
induced character Ind
of H. To see this, we first observe that
L0 (ϑτ ) is a supercharacter
⊥
−1
−1
−1
L = ϑ (ϑH) = ϑ (ϑh) : h ∈ H where ϑ is the inverse of ϑ in the group A◦ ;
recall that ϑ−1 (a) = ϑ(−a) for all a ∈ A. Indeed, we have
(ϑ−1 (ϑh))(a) = ϑ−1 (a)(ϑh)(a) = ϑ(−a)ϑ(ah−1 ) = ϑ(a(h−1 − 1)) = 1
for all h ∈ H and all a ∈ L, and hence ϑ−1 (ϑH) ⊆ L⊥ ; for the reverse inclusion, we
note that |L⊥ | = |B|/|L| = |H|/|L| = |Hϑ| = |ϑH| = |ϑ−1 (ϑH)|. It follows that there
are h1 , . . . , hq ∈ H such that τi = ϑ−1 (ϑhi ) for all 1 ≤ i ≤ q. Since ϑ0 ϑ−1 = 1B , we see
that ϑ0 τi = ϑ0 hi for all 1 ≤ i ≤ q, and so
H d
ζϑ = IndH (ϑd
0 h1 ) + · · · + Ind (ϑ0 hq ).
L0
L0
Finally, for every 1 ≤ i ≤ q, it is clear that L0 is the left centraliser of ϑ0 hi in H, and
d
thus by the definition IndH
L0 (ϑ0 hi ) is the supercharacter χϑ0 hi of H. Moreover, since
ϑ0 hi ∈ Hϑ0 H, we have χϑ0 hi = χϑ0 for all 1 ≤ i ≤ q, and so ζϑ = qχϑ0 .
Finally, replacing ϑ by gϑ in the argument above, we get ζgϑ = qξgϑ0 for all g ∈ G,
and the result follows from Proposition 2.17.
We are now ready to prove the main result of this section.
Theorem 2.21. Let G = 1+A be an algebra group over the field k, let B be an arbitrary
subalgebra of A, and let H = 1 + B. Then, the restriction to H of any supercharacter
of G is a nonnegative integer linear combination of supercharacters of H.
Proof. It is enough to choose a maximal subalgebra of A containing B and to proceed
by induction on |G| = |A|.
SUPERCHARACTERS AND SET PARTITION COMBINATORICS
27
Next, we begin working towards a proof that the product of two supercharacters of
an algebra group is always a nonnegative integer linear combination of supercharacters.
Let A and B be finite-dimensional nilpotent k-algebras, and consider the algebra groups
G = 1 + A and H = 1 + B. Then, the cartesian product A ⊕ B is also a nilpotent
k-algebra for the product defined by (a, b) · (a′ , b′ ) = (aa′ , bb′ ) for all a, a′ ∈ A and all
b, b′ ∈ B, and the the algebra group 1 + (A × B) is canonically isomorphic to the direct
product G × H. If χ and χ′ are characters of H and K respectively, then we define the
function χ × χ′ : G × H → C by the rule (χ × χ′ )(g, h) = χ(g)χ′ (h) for all g ∈ G and all
h ∈ H, and it is a standard fact that χ × χ′ is a character of G × H (see the exercises).
Analogously, if ϑ ∈ A◦ and ϑ′ ∈ B◦ , we can define the linear character ϑ × ϑ′ of A × B
by (ϑ × ϑ′ )(a, b) = ϑ(a)ϑ′ (b) for all a ∈ A and all b ∈ B. Furthermore, it is not hard to
see that every linear character of A × B is uniquely of the form ϑ × ϑ′ for some ϑ ∈ A◦
and ϑ′ ∈ B◦ , and thus that the dual group (A × B)◦ can be identified with the direct
product A◦ × B◦ .
Lemma 2.22. In the notation as above, let χϑ be the supercharacter of G associated
with ϑ ∈ A◦ , and let χϑ′ be the supercharacter of H associated with ϑ′ ∈ B◦ . Then,
χϑ × χϑ′ is the supercharacter χϑ×ϑ′ of G × H associated with ϑ × ϑ′ ∈ (A × B)◦ .
Proof. By definition, χϑ is afforded by the CH-submodule Lϑ = CG·εϑ of CA, whereas
χϑ′ is afforded by the CH-submodule Lβ = CH · εϑ′ of CB. Then, χϑ × χϑ′ is the
character of G × H afforded by the C(G × H)-module Lϑ ⊗ Lϑ′ where the (G × H)action is given by (g, h) · (u ⊗ v) = (g · u) ⊗ (h · v) for all g ∈ G, h ∈ H, u ∈ Lϑ
and v ∈ Lϑ′ . It is routine to check there is an isomorphism of C(G × H)-modules
φ : C(A × B) → CA ⊗ CB defined by φ(a, b) = a ⊗ b for all a ∈ A and b ∈ B. We have
φ(εϑ×ϑ′ ) = εϑ ⊗ εϑ′ , and thus φ(Lϑ×ϑ′ ) = Lϑ ⊗ Lϑ′ . The result follows.
We are now able to prove the following.
Theorem 2.23. Let G = 1 + A be an algebra group over k, and let χ and χ′ be
supercharacters of G. Then, χχ′ is a nonnegative integer linear combination of supercharacters of G.
Proof. By the previous lemma, the function
χ × χ′ is a supercharacter of the algebra
group G × G = 1 + (A × A). Let D = (a, a) : a ∈ A be the diagonal subalgebra of
A×A, and note that there is a natural isomorphism of k-algebras D ∼
= A. Let D = 1+D
be the corresponding algebra subgroup of G×G, and observe that D = (g, g) : g ∈ G
is the diagonal subgroup of G × G. It is obvious that there is a natural isomorphism of
groups D ∼
= G, which allows us to identify D with G. Then, we see that the restriction
ResG×G
(χ
× χ′ ) is precisely the usual product of characters χχ′ . Since D = 1 + D is
D
an algebra subgroup of G × G, Theorem 2.21 implies that χχ′ is a nonnegative integer
linear combination of supercharacters of D, and hence of G.
Finally, we discuss induction of supercharacters. We have shown that the restriction
of a supercharacter of an algebra group to an algebra subgroup is always a nonnegative
integer linear combination of supercharacters, and so it is tempting to guess that the
analogous property also holds for induction. However, the character induced by a
supercharacter of an algebra subgroup need not even be a superclass function (in other
words, it may not be constant on superclasses). However, it is possible to modify the
definition of induction in algebra groups so that the result of “inducing” a superclass
function of an algebra subgroup is always a superclass function. Furthermore, this
can be done so that the analog of Frobenius reciprocity is valid. But, even with this
28
CARLOS A. M. ANDRÉ
modified induction, it is not generally true that supercharacters of an algebra subgroup
yield integer linear combinations of supercharacters of the whole group.
Let G = 1 + A be an algebra group over k, and let H = 1 + B be an algebra
subgroup of G. If φ is an arbitrary complex-valued function of H, we define the
superinduced function SIndG
H (φ) on G by the formula
X
1
(2d)
SIndG
φ◦ (1 + h(g − 1)k)
H (φ)(g) =
|G| |H| h,k∈G
for all g ∈ G. Even with no assumption on φ, it is immediate that SIndG
H (φ) is a
superclass function on G. Observe that
SIndG
H (φ)(1) =
|G|2
φ(1) = |G : H| φ(1).
|G| |H|
The multiplicative constant 1/(|G||H|) in the definition is chosen to guarantee that the
following analog of Frobenius reciprocity holds.
Theorem 2.24. Let G = 1 + A be an algebra group over k, and let H = 1 + B be
an algebra subgroup of G. Let φ be a complex-valued function of H, and let ψ be a
superclass function of G. Then,
G
SIndG
H (φ), ψ G = φ, ResH (ψ) H .
Proof. We have
SIndG
H (φ), ψ
G
1 X
SIndG
H (φ)(g) ψ(g)
|G| g∈G
X
1
φ◦ (1 + hak) ψ(1 + a).
=
|G|2 |H| a∈A
=
h,k∈G
Since ψ is a superclass function, we have ψ(1 + a) = ψ(1 + hak) for all a ∈ A and all
h, k ∈ G, and thus
1 X ◦
(φ),
ψ
=
SIndG
φ (1 + a) ψ(1 + a)
H
G
|H| a∈A
1 X
φ(1 + a) ψ(1 + a) = φ, ResG
(ξ)
=
H
H
|H| a∈B
as required.
Exercises.
2.1. Let G be a finite group and suppose that G acts on the left and right of some set
Ω and that the two actions commute. Prove that for every ω ∈ Ω
|GωG| =
|Gω||ωG|
.
|Gω ∩ ωG|
Then, conclude that, if G is an algebra group over k = Fq , then the superclasses of G all
have q-power size. [Hint. Observe that GωG is a union of left orbits that are transitively
permuted by the right action of G, and the right action of G is also transitive on the
set of intersections of ωG with the right translates of Gω.]
SUPERCHARACTERS AND SET PARTITION COMBINATORICS
29
2.2. Let G = 1 + A be an algebra group over k. Prove that the following hold.
(1) If a ∈ A is such that L(a) + R(a) = A, then the superclass of 1 + a is the
conjugacy class of 1 + a.
(2) If ϑ ∈ A◦ is such that L(ϑ) + R(ϑ) = A, then GϑG is the conjugation orbit of
ϑ.
2.3. Let G = 1 + A be an algebra group over k, and let ϑ ∈ A◦ . Prove that the
following hold.
(1) Gϑ = ϑ · R(ϑ)⊥ = ϑτ : τ ∈ R(ϑ)⊥ .
(2) ϑG = ϑ · L(ϑ)⊥ = ϑσ : σ ∈ L(ϑ)⊥ .
(3) |Gϑ| = |ϑG| and |L(ϑ)| = |R(ϑ)|.
Moreover, conclude that the left, right and two-sided orbits of G on A◦ have q-power
size where q = |k|.
2.4. Let G = 1 + A be an algebra group over k, and let ϑ ∈ A◦ . Prove that the
following hold.
(1) The supercharacter χϑ is irreducible if and only if L(ϑ) + R(ϑ) = A.
(2) If the supercharacter χϑ has a linear constituent, then Gϑ = ϑG = GϑG, and
R(ϑ) = L(ϑ) is an ideal of A.
2.5. Let G = 1 + A be an algebra group over k, let ϑ ∈ A◦ , and let χ ∈ irr(G) be an
irreducible constituent of χϑ . Prove that
χ(1)|Gλ|
χ, χλ =
,
|GϑG|
and conclude that
X
|GϑG| =
χ(1)2 .
χ∈irrϑ (G)
2.6. ([10, Section 4]) Let G = 1 + A be an algebra group over k, let ϑ ∈ A◦ , and let
S(ϑ) = g ∈ G : gϑ = ϑh for some h ∈ G .
Prove that the following hold.
(1) L(ϑ) is a normal subgroup of S(ϑ), and the linear character ϑ̂ : L(ϑ) → C is
S(ϑ)-invariant (that is, ϑ̂(ghg −1) = ϑ̂(h) for all g ∈ S(ϑ) and all h ∈ L(ϑ)).
S(ϑ)
(2) If irr(S(ϑ); ϑ̂) denotes the set of all the irreducible constituents of IndL(ϑ) (ϑ̂),
then
X
S(ϑ)
S(ϑ)
ResL(ϑ) (φ) = φ(1)ϑ̂ and IndL(ϑ) (ϑ̂) =
φ(1)φ.
φ∈irr(S(ϑ);ϑ̂)
(3) The supercharacter χϑ decomposes as the sum
X
χϑ =
φ(1) IndG
S(ϑ) (φ).
φ∈irr(S(ϑ);ϑ̂)
(4) The mapping φ 7→ IndG
L(ϑ) (φ) defines a bijection from irr(S(ϑ); ϑ̂) to irr(G; ϑ̂)
which satisfies
L(ϑ)
χϑ , IndG
S(ϑ) (φ) G = IndS(ϑ) (ϑ̂), φ S(ϑ) = φ(1)
for all φ ∈ irr(S(ϑ); ϑ̂). [Hint. Observe that |S(ϑ) : L(ϑ)| = χϑ , χϑ . Choose
an irreducible constituent of IndG
L(ϑ) (φ) and analyse its multiplicity.]
30
CARLOS A. M. ANDRÉ
2.7. ([18]) Let A be the nilpotent k-algebra with basis {a, b, c} where ab = c and all
other products of basis vectors are zero, and consider the algebra group G = 1 + A.
(1) Prove that H = 1 + ka is an algebra subgroup of G, and that {1 + a} is a
superclass of H.
(2) Define φ : H → C by setting φ(1 + a) = 1, and φ(h) = 0 for all other elements
h ∈ H. Show that φ is a superclass function of H.
(3) Prove that 1+a is the only element of G on which the induced function IndG
H (φ)
is nonzero. [Notice that G is abelian.]
(4) Observe that 1 + c and 1 + x are in the same superclass of G, and conclude that
IndG
H (φ) is not a superclass function.
2.8. ([18]) Let F2 be the field with 2 elements, let A be the nilpotent F2 -algebra with
basis {a, b, c, d} where a2 = b + c + d, ba = ac = d and all other products of basis
vectors are zero, and consider the algebra group G = 1 + A.
(1) Prove that, if B = F2 b + F2 c + F2 d, then B2 = 0, H = 1 + B is an abelian
algebra subgroup of G, and all supercharacters of H are linear.
(2) Note that SIndG
H (χ)(1) = 2 for all supercharacter χ of H.
(3) Let ϑ ∈ A◦ be such that ker(ϑ) = B. Prove that R(ϑ) = kb + kd, and conclude
that the supercharacter χϑ has degree 4.
(4) Prove that, if a supercharacter χ, ResG
H (χϑ ) 6= 0 for some supercharacter χ of
H, then χϑ occurs in SIndG
H (χ) with a positive non-integral coefficient.
3. Supercharacters and superclasses of the unitriangular group
Throughout this section, let k be a finite field with q elements, let n ∈ N, and
denote by Un the (upper) unitriangular group consisting of all upper-triangular n × n
matrices with coefficients in k and ones on the diagonal. Then, Un = 1 + un where
1 = 1n denotes the identity matrix and un denotes the k-algebra consisting of all
strictly upper-triangular n × n matrices with coefficients in k. Since un is a finitedimensional nilpotent k-algebra, the unitriangular group Unis an algebra group over
k. Furthermore, we define [n] = {1, 2, . . . , n} and [[n]] = (i, j) : 1 ≤ i < j ≤ n .
For every (i, j) ∈ [[n]] we denote by ei,j ∈ un the matrix
with 1 in the (i,
j)th entry
and zeroes elsewhere, and consider the standard k-basis ei,j : (i, j) ∈ [[n]] ; hence, for
every a ∈ un , we have
X
a=
ai,j ei,j
(i,j)∈[[n]]
where ai,j ∈ k is the (i, j)th coefficient of the matrix a.
3.1. Set partitions. A set partition π of [n] is a family of nonempty sets, called blocks,
whose disjoint union is [n]; we define the length ℓ(π) of π to be the number of blocks,
and we write π = B1 /B2 / . . . /Bℓ ⊢ [n] where B1 , B2 , . . . , Bℓ are the blocks of π are the
blocks of π which we agree to be ordered by increasing value of the smallest element in
the block (this implied order will allow us to reference the ith block of the set partition
without ambiguity). If π ⊢ [n] and (i, j) ∈ [[n]], then the pair (i, j) is said to be an arc
of π if i and j occur in the same block B of π and there is no k ∈ B with i < k < j; we
denote by D(π) the set consisting of all arcs of π. Then, the standard representation of
π ⊢ [n] is the directed graph with vertex set [n] and edge set D(π) drawn by listing the
elements of [n] in the natural order with the corresponding arcs overhead; notice that
the connected components of this graph are precisely the blocks of the set partition π.
SUPERCHARACTERS AND SET PARTITION COMBINATORICS
31
As an example, π = 157/3/4/689 is a set partition of [9] with blocks {1, 5, 7}, {3}, {4}
and {6, 8, 9}; the standard representation of π is
b
b
b
b
b
b
b
b
b
1
2
3
4
5
6
7
8
9
Every set partition of [n] is uniquely determined by its set of arcs, and hence by a
subset of [[n]]; obviously not every subset of [[n]] is the set of arcs of a set partition of
[n]. We say that D ⊆ [[n]] is a basic subset if D = D(π) for some π ⊢ [n]; the basic
subsets of [[n]] may be characterised without reference to set partitions. In fact, the set
[[n]] can be naturally identified with the set of entries of a (strictly) upper-triangular
matrix of size n × n. Thus,
it is natural to define for each 1 ≤ i ≤ n the ith row of [[n]]
to be the subset ri (n)
=
(i, j) : i < j ≤ n , and for each 1 ≤ j ≤ n the jth column of
[[n]] to be cj (n) = (i, j) : 1 ≤ i < j . Then, D ⊆ [[n]] is a basic subset if and only if
it contains at most one entry from each row and at most one entry from each column.
From this, we see that the set partitions of [n] are in one-to-one correspondence with
the basic subsets of [[n]]; notice that the empty subset ∅ ⊆ [[n]] is basic, and that it
corresponds to the set partition π = 1/2/ . . . /n of [n].
There are some natural statistics on set partitions which will be of interest to us.
Firstly, for every π ⊢ [n], we define the dimension dim(π) of π to be the sum
X
dim(π) =
# k ∈ [n] : i < k ≤ j .
(i,j)∈D(π)
In a certain sense dim(π) measures how far the set ofarcs D(π) is from the empty
subset of [[n]] (or, in matrix terms, from the diagonal (i, i) : i ∈ [n] . For example,
if D(π) = {(i, j)} is a singleton set, then dim(π) = j − i. For simplicity, for all
1 ≤ i < j ≤ n we write {ij} (or {ij}n if it necessary to emphasise that {ij} is a set
partition of [n]) to denote the unique set partition π ⊢ [n] such that D(π) = {(i, j)};
we also write dim(i, j) instead of dim({ij}). It is obvious that
X
dim(π) =
dim(i, j)
(i,j)∈D(π)
for all π ⊢ [n].
On the other hand, we define a crossing of π ⊢ [n] to be a unordered pair of arcs
{(i, j), (k, l)} ⊆ D(π) such that i < k < j < l; we denote by crs(π) the number of
crossings of π, and refer to crs(π) as the crossing number of π. Diagrammatically, a
crossing of π corresponds to the picture
b
b
b
b
i
k
j
l
We say that π is a non-crossing set partition if π has no crossings, that is, if crs(π) =
0. Notice that a crossing {(i, j), (k, l)} of π corresponds to the entry (k, j) in the
intersection rk (n) ∩ cj (n) of the kth row and the jth column of [[n]]; of course, this
intersection is nonempty if and only if 1 < k < j < l.
The third statistic is the nesting number nst(π) of π ⊢ [n] which is defined as the
number of nests of π; a nest of π we mean a unordered pair of arcs {(i, j), (k, l)} ⊆ D(π)
such that i < k < l < j. Diagrammatically, a nest of π corresponds to
32
CARLOS A. M. ANDRÉ
b
b
b
b
i
k
l
j
We say that π is a non-nesting set partition if π has no nests, that is, if nst(π) = 0.
To illustrate, if π = 159/2367/48 ⊢ [9], then dim(π) = 17, crs(π) = 4, and nst(π) =
3, as is easily observed in the diagrammatic representation:
b
b
b
b
b
b
b
b
b
1
2
3
4
5
6
7
8
9
For our purposes it also convenient to introduce the statistic nstπ (σ) on a pair of set
partitions π, σ ⊢ n to be the sum
X
nstπ (σ) =
nsti,j (σ)
(i,j)∈D(π)
where for any (i, j) ∈ [[n]] we define
nsti,j (σ) = # (k, l) ∈ D(σ) : i < k < l < j .
In other words, ni,j (σ) is the number of arcs of σ which are strictly covered by (i, j);
we say that (i, j) covers (k, l) if i ≤ k < l ≤ j, and we say that (i, j) strictly covers
(k, l) if i < k < l < j. Thus, if we denote by Di,j (σ) the set of all arcs (k, l) ∈ D(σ)
which are strictly covered by (i, j), then nsti,j (σ) = #Di,j (σ); notice that
X
nstπ (π) =
nsti,j (π) = nst(π)
(i,j)∈D(π)
is precisely the nesting number of π.
3.2. k× -coloured set partitions. By an k× -colouring of a set partition π ⊢ [n] we
mean a map φ : D(π) → k× ; recall that k× = k \ {0}. An k× -coloured set partition
of [n] is then a pair (π, φ) where π ⊢ [n] and φ is an k× -colouring of π; we denote by
Sn (k) the set consisting of all k× -coloured set partitions of [n]. If (π, φ) ∈ Sn (k), then
an arc of (π, φ) is formally defined to be a pair ((i, j), α) ∈ [[n]] × k where (i, j) ∈ D(π)
and α = φ(i, j) is the colour of (i, j); we denote by D(π, φ) the set consisting of all
coloured arcs of (π, φ). The standard representation of (π, φ) ∈ Sn (k) is the directed
k× -coloured graph with vertex set [n] and edge set D(π) where we give the colour φ(i, j)
to the edge (i, j). For example,
α
γ
β
α
γ
b
b
b
b
b
b
b
b
b
1
2
3
4
5
6
7
8
9
is the standard representation of (π, φ) ∈ S9 (k) where π = 157/23/4/689 and φ(1, 5) =
φ(6, 8) = α, φ(5, 7) = β and φ(2, 3) = φ(8, 9) = γ for α, β, γ ∈ k× .
With every (π, φ) ∈ Sn (k) we associate the matrix
X
(3a)
eπ,φ =
φ(i, j)ei,j ;
(i,j)∈D(π)
the matrix eπ,φ may be thought as an encoding of the k× -coloured set partition (π, φ),
and vice-versa. On the other hand, we also note that the set Sn (k) can be naturally
identified with the set consisting of all pairs (D, φ) where D is a basic subset of [[n]]
SUPERCHARACTERS AND SET PARTITION COMBINATORICS
33
and φ : D → k× is any map; we refer to the map φ as an k× -colouring of D, and to
(D, φ) as a k× -coloured basic subset of [[n]].
3.3. Superclasses of Un . We recall from Subsection 2.2 that the superclasses of Un
correspond to the two-sided orbits of Un on un under the mapping Un aUn 7−→ 1+Un aUn
for a ∈ un . Since every two-sided orbit of Un on un contains a unique matrix of the
form eπ,φ for some (π, φ) ∈ Sn (k) (obtained by a set of elementary row and column
operations), the superclasses of Un are parametrised by k× -coloured set partitions of
[n]. For every (π, φ) ∈ Sn (k), we set
Oπ,φ = Un eπ,φ Un
and Kπ,φ = 1 + Oπ,φ ;
in the case where D(π) = {(i, j)} and φ(i, j) = α, we simplify the notation and write
O{ij},α and K{ij},α instead of Oπ,φ and Kπ,φ , respectively. In this special situation, then
it is not hard to verify that K{ij},α consists of all matrices g ∈ Un satisfying


if, either 1 ≤ k < j, or i < l ≤ n,
0,
gk,l = α,
if k = i and l = j,

α−1 g g , if k < i < j < l.
k,j i,l
In the general situation, we have the following result.
Theorem 3.1. If (π, φ) ∈ Sn (k), then the superclass Kπ,φ can factorises (uniquely) as
the product
Y
(3b)
Kπ,φ =
K{ij},φ(i,j).
(i,j)∈D(π)
Similarly, the two-sided orbit Oπ,φ decomposes (uniquely) as the sum
X
(3c)
Oπ,φ =
O{ij},φ(i,j) .
(i,j)∈D(π)
Proof. Since g(1 + a) = g + ga for all g ∈ Un and a ∈ un , it is clear that (3c) implies
(3b). Let (π ′ , φ′) ∈ Sn (k) be the k× -coloured set partition obtained from (π, φ) by
removing an arc ((i, j), φ(i, j)) ∈ D(π, φ). By induction, we assume that
X
O{kl},φ(k,l) ,
Oπ′ ,φ′ =
(k,l)∈D(π ′ )
and the result will follow once we prove that Oπ,φ = Oπ′ ,φ′ + O{ij},φ(i,j). Since Oπ′ ,φ′ +
O{ij},φ(i,j) is two-sided invariant and eπ,φ = eπ′ ,φ′ + φ(i, j)ei,j ∈ Oπ′ ,φ′ + O{ij},φ(i,j) , we
have Oπ,φ ⊆ Oπ′ ,φ′ + O{ij},φ(i,j) . Conversely, it is clear that every element of Oπ′ ,φ′ +
O{ij},φ(i,j) lies in the two-sided orbit of eπ′ ,φ′ + a for some a ∈ O{ij},φ(i,j) . Since D(π)
is a basic subset of [[n]], we may obtain φ(i, j)ei,j from a using elementary row and
column operations which do not change eπ′ ,φ′ . This shows that every element of Oπ′ ,φ′ +
O{ij},φ(i,j) lies in Oπ,φ as required.
Remark 3.2. By Exercise 2.2, we know that for every k× -coloured arc ((i, j), α) the
two-sided orbit O{ij},α is a conjugation orbit of un , and hence the superclass K{ij},α is a
conjugacy class of Un . Thus, the above factorisation means that every superclass of Un
is a product of elementary conjugacy classes where we define an elementary conjugacy
class of Un to be the conjugacy class which contains an element of the form αei,j for
α ∈ k× and (i, j) ∈ [[n]].
34
CARLOS A. M. ANDRÉ
3.4. Supercharacters. By the definition, the supercharacters of Un correspond to the
two-sided orbits Un ϑUn of the dual group u◦n of u+
n under the mapping
X
|Un ϑ|
τ̂
Un ϑUn 7−→
|Un ϑUn | τ ∈U ϑU
n
u◦n ;
n
u◦n
for ϑ ∈
we recall that τ̂ (g) = τ (g − 1) for all τ ∈
and all g ∈ Un . In this section,
we associate a supercharacter χπ,φ with every k× -coloured set partition (π, φ) ∈ Sn (k);
we recall that the number of supercharacters is equal to the number of superclasses,
and hence to the number of k× -coloured set partitions.
To start with, we describe the linear characters of the additive group u+
n of un . We
define an k-bilinear form on un by the rule
a · b = Tr(aT b)
for all a, b ∈ un . Notice that the standard k-basis ei,j : (i, j) ∈ [[n]] is orthonormal with respect to this bilinear form (which means that ei,j · ek,l = δi,k δj,l for all
(i, j), (k, l) ∈ [[n]]), and thus the bilinear form identifies the vector space un with its
dual u∗n = Homk (un , k). In particular, we see that the form is non-degenerated in the
⊥
sense that u⊥
n = 0; as usual, for every vectorsubspace v of un , we write v to denote
the orthogonal subspace of v, that is, v⊥ = a ∈ un : a · b = 0 for all b ∈ v . On the
other hand, we fix a nontrivial linear character ϑ : k+ → C× and for every a ∈ un we
define the map ϑa : un → C× by
(3d)
ϑa (b) = ϑ(a · b)
for all b ∈ un .
+
◦
Lemma 3.3. The
mapping a 7→ ϑa defines an isomorphism of groups Θ : un → un . In
◦
particular, un = ϑa : a ∈ un .
Proof. It is routine to check that ϑa ∈ u◦n for all a ∈ un , and that ϑa+b = ϑa ϑb for all
a, b ∈ un . Thus, Θ is a group homomorphism with kernel
ker(Θ) = a ∈ un : ϑ(a · b) = ϑa (b) = 1 for all b ∈ un .
Since a · (αei,j ) = αai,j for all α ∈ k and all (i, j) ∈ [[n]], we see that a ∈ ker(Θ) if and
only if ϑ(αai,j ) = 1 for all α ∈ k and all (i, j) ∈ [[n]]. Therefore, if ai,j 6= 0 for some
(i, j) ∈ [[n]], then we deduce that
1 X
1 X
1=
ϑ(αai,j ) =
ϑ(β) = ϑ, 1k ,
|k| α∈k
|k| β∈k
and this contradicts the non-triviality of ϑ. It follows that ker(Θ) = 0, and hence Θ is
injective. It is surjective because |u◦n | = |un |.
For every (π, φ) ∈ Sn (k), we define ϑπ,φ to be the linear character ϑeπ,φ ∈ u◦n which
corresponds to eπ,φ ∈ un ; hence, if we write ai,j for the (i, j)th-coefficient ai,j = a · ei,j
of a ∈ un , then
Y
Y
(3e)
ϑπ,φ (a) = ϑ(a · eπ,φ ) =
ϑ(φ(i, j)(a · ei,j )) =
ϑ(φ(i, j)ai,j )
(i,j)∈D(π)
(i,j)∈D(π)
for all a ∈ un . Thus, if we define ϑ{ij},α = ϑαei,j for all (i, j) ∈ [[n]] and all α ∈ k× ,
then we obtain a factorisation
Y
(3f)
ϑπ,φ =
ϑ{ij},φ(i,j)
(i,j)∈D(π)
SUPERCHARACTERS AND SET PARTITION COMBINATORICS
35
as a product of the linear characters which correspond to the arcs of (π, φ). On the
other hand, we define χπ,φ to be the supercharacter of Un corresponding to the twosided orbit Un ϑπ,φ Un of ϑπ,φ ∈ u◦n . Similarly, for all (i, j) ∈ [[n]] and all α ∈ k× ,
we define χ{ij},α to be the supercharacter of Un corresponding to the two-sided orbit
Un ϑ{ij},α Un of ϑ{ij},α ∈ u◦n , and refer to these as the elementary characters of un .
Theorem 3.4. If X = Kπ,φ : (π, φ) ∈ Sn (k) and Y = χπ,φ : (π, φ) ∈ Sn (k) , then
the pair (X , Y) is a supercharacter theory of Un .
Proof. We know from Subsection 2.2 that X is the set of superclasses of Un . On
the other hand, X consists of supercharacters of G, and thus it is enough to prove
that |Y| = |X | (because the total number of supercharacters equals the number of
superclasses). Since supercharacters are in one-to-one correspondence with two-sided
orbits of Un on u◦n , it is enough to prove that every linear character ϑ ∈ u◦n lies in the
two-sided orbit of ϑπ,φ for some (π, φ) ∈ Sn (k). By the previous lemma, we know that
every linear character of u◦n is of the form ϑa for some a ∈ un . We have
ϑa (gbh) = ϑ(Tr(aT gbh) = ϑ(Tr(haT gb) = ϑ(Tr((g T ahT )T b))
for all b ∈ un and all g, h ∈ Un . It is clear that there exist g, h ∈ Un such that
g T ahT = eπ,φ + u for a unique (π, φ) ∈ Sn (k) and some lower triangular matrix u. Since
Tr((eπ,φ + u)T b) = Tr((eπ,φ )T b), we conclude that
(g −1ϑa h−1 )(b) = ϑa (gbh) = ϑπ,φ (b)
for all b ∈ un , and hence the linear character ϑa ∈ u◦n lies in the two-sided orbit of ϑπ,φ .
The claim follows.
We recall from Subsection 2.4 that, if επ,φ denotes the idempotent εϑπ,φ of Cun
and Lπ,φ denotes the CUn -submodule CG · επ,φ of Cun , then χπ,φ is the character
of Un afforded by Lπ,φ . In particular, letting ε{ij},α = εϑ{ij},α for all (i, j) ∈ [[n]]
and α ∈ k× , we see that χ{ij},α is the character of Un afforded
by the CUn -module
L{ij},α = CG · ε{ij},α . We also recall that Lπ,φ has C-basis εϑ : ϑ ∈ Un ϑπ,φ ; in
particular, L{ij},α has C-basis εϑ : ϑ ∈ Un ϑ{ij},α . The proof of the the following
result is straightforward, and is left as an exercise for the reader.
Lemma 3.5. If (i, j) ∈ [[n]] and α ∈ k× , then
Un ϑ{ij},α = ϑa : a ∈ αei,j + Vi,j
where Vi,j denotes the vector subspace
P of un spanned by {ei,i+1 , . . . , ei,j−1}. More generally, if (π, φ) ∈ Sn (k) and Vπ = (i,j)∈D(π) Vi,j , then
Un ϑπ,φ = ϑa : a ∈ eπ,φ + Vπ .
As a consequence, we are now able to prove that for every (π, φ) ∈ Sn (k) the supercharacter χπ,φ factorises as the product of the elementary characters corresponding
to the arcs of
(π, φ). Firstly, we recall that the group algebra Cun has a C-basis
◦
for every
εϑ : ϑ ∈ un ; henceforth, we simplify the notation, and define εa = εϑa a ∈ u◦n ; then, in virtue of the isomorphism Θ : un → u◦n , the set εa : a ∈ un is a Cbasis of Cun . For every t ∈ N, we consider the tensor power (Cun )⊗t = Cun ⊗ · · · ⊗ Cun
(t factors) endowed with the natural structure of CUn -module. As a vector space,
(Cun )⊗t has a C-basis consisting of all pure tensors εa1 ⊗ · · · ⊗ εat for a1 , . . . , at ∈ un ,
and it is routine to check that
g · (εa1 ⊗ · · · ⊗ εat ) = ϑa1 +···+at (g −1 − 1)(εgϑa1 ⊗ · · · ⊗ εgϑat )
36
CARLOS A. M. ANDRÉ
for all g ∈ Un and all a1 , . . . , at ∈ un ; notice that ϑa ϑb = ϑa+b for all a, b ∈ un . Then,
we can define a C-linear map η : (Cun )⊗t → Cun by setting
η(εa1 ⊗ · · · ⊗ εat ) = εa1 +···+at
for all a1 , . . . , at ∈ un , and extending by linearity. Since gϑa+b = g(ϑa ϑb ) = (gϑa )(gϑb )
for all g ∈ Un and all a, b ∈ un , it is easily seen that
η(g · (εa1 ⊗ · · · ⊗ εat )) = g · η(εa1 ⊗ · · · ⊗ εat )
for all g ∈ Un and all a1 , . . . , at ∈ un , and this implies that η is in fact a homomorphism
of CUn -modules.
Theorem 3.6. Let (π, φ) ∈ Sn (k), and let D(π) = {(i1 , j1 ), . . . , (it , jt )}. Then, there
is an isomorphism of CUn -modules
Lπ,φ ∼
= L{i1 j1 },φ(i1 ,j1 ) ⊗ · · · ⊗ L{it jt },φ(it ,jt ) .
In particular, the supercharacter χπ,φ factorises as the product
Y
(3g)
χπ,φ =
χ{ij},φ(i,j)
(i,j)∈D(π)
of the elementary characters corresponding to the arcs of (π, φ).
r jr },φ(ir ,jr )
Proof. For simplicity, we set Lr = L{i
for
1 ≤ r ≤ t. By the previous lemma,
it follows that each Lr has C-basis εa :a ∈ Vr where Vr = φ(ir , jr )e(ir ,jr ) + V(ir ,jr ) ,
and thus L1 ⊗ · · · ⊗ Lt has a C-basis εa1 ⊗ · · · ⊗ εat : ar ∈ Vr , 1 ≤ r ≤ t . On
the other hand, the previous lemma also implies that Lπ,φ has C-basis εa : a ∈ V
where V = V1 + · · · + Vt . Since the mapping (a1 , . . . , at ) 7→ a1 + · · · + at is easily
seen to define a bijection V1 × · · · × Vt → V, we conclude that η : (Cun )⊗t → Cun
restricts to an isomorphism of CUn -modules η : L1 ⊗ · · · ⊗ Lt → Lπ,φ . In particular,
the CUn -modules Lπ,φ and L1 ⊗ · · · ⊗ Lt afford the same character of Un , and hence
χπ,φ = χ{i1 j1 },φ(i1 ,j1 ) · · · χ{it jt },φ(it ,jt ) . The proof is complete.
3.5. Supercharacter values. We now determine the value of supercharacters on superclasses. Let (π, φ), (σ, ψ) ∈ Sn (k) be arbitrary, and recall that Kσ,ψ is the superclass
which contains the matrix gσ,ψ = 1 + eσ,ψ ∈ Un . Then, χπ,φ is constant on Kσ,ψ , our
goal is to evaluate χπ,φ (gσ,ψ ). By Theorem 3.6, we know that
Y
χπ,φ (gσ,ψ ) =
χ{ij},φ(i,j)(gσ,ψ ),
(i,j)∈D(π)
and thus it is enough to evaluate χ{ij},α (gσ,ψ ) for all (i, j) ∈ [[n]] and all α ∈ k× . By
Theorem 2.12, we have
X
|Un ϑ{ij},α |
χ{ij},α (gσ,ψ ) =
τ (eσ,ψ ).
|Un ϑ{ij},α Un |
{ij},α
τ ∈Un ϑ
Un
As seen in the proof of Theorem 3.4,
Un ϑ{ij},α Un = ϑ(gT (αei,j )hT )sup : g, h ∈ Un
where asup denotes the upper-triangular part of a matrix a. From this it it easily
verified that
Un ϑ{ij},α Un = ϑa : a ∈ O{ij},α
SUPERCHARACTERS AND SET PARTITION COMBINATORICS
37
where O{ij},α is the set of all matrices a ∈ un satisfying


if, either 1 ≤ k < i, or j < l ≤ n,
0,
(3h)
ak,l = α,
if (k, l) = (i, j),

α−1 a a , if i < k < l < j;
k,j i,l
hence |Un ϑ{ij},α Un | = q 2(j−i−1) . It follows that
X
1
1
τ (eσ,ψ ) = j−i−1
χ{ij},α (gσ,ψ ) = j−i−1
q
q
{ij},α
τ ∈Un ϑ
Un
X
ϑa (eσ,ψ );
a∈O {ij},α
we recall from Lemma 3.10 and Lemma 2.8 that |Un ϑ{ij},α | = |ϑ{ij},α Un |. Moreover,
we have
Y
Y
ϑ(ψ(k, l)α−1 ar,j ai,s )
ϑa (eσ,ψ ) = ϑ(Tr(aT eσ,ψ )) =
ϑ(ψ(k, l)ar,s ) =
(k,l)∈D(σ)
(k,l)∈D(σ)
i≤k<l≤j
for all a ∈ O{ij},α , and thus
X
XX
(3i)
ϑa (eσ,ψ ) =
Y
ϑ(ψ(k, l)ul−i vk−i+1 )
u∈U v∈V (k,l)∈D(σ)
i≤k<l≤j
a∈O {ij},α
where U = u ∈ k j−i : uj−i = c and V = v ∈ k j−i : v1 = c .
Now, suppose that (k, j) ∈ D(σ) for some i < k < j. Then, the right hand side of
(3c) becomes
!
!
X
X X
Y
ϑ(ψ(k ′ , l′ )ul′ −i vk′ −i+1 ) .
ϑ(αψ(k, j)v)
v∈k
u∈U
v∈V
(k ′ ,l′ )∈D(σ)
vk−i+1 =0 i≤k ′ <l′ <j
Since the mapping v 7→ ϑ(αψ(k, j)v) defines a nontrivial character of the additive
group of k, we have
X
ϑ(αψ(k, j)v) = 0,
v∈k
and this implies that χ{ij},α (gσ,ψ ) = 0. Similarly, if (i, k) ∈ D(σ) for some i < k < j,
then χ{ij},α (gσ,ψ ) = 0.
On the other hand, suppose that (i, k), (k, j) ∈
/ D(σ) for all i < k < j, and let
Di,j (σ) = {(k1 , l1 ), . . . , (kt , lt )}
where k1 < . . . < kt ; recall that Di,j (σ) = (k, l) ∈ D(σ) : i < k < l < j , and that
|Di,j (σ)| = nsti,j (σ). Then, the right hand side of (3c) can be rewritten as
X
X Y
q 2(j−i−1−t) ϑ(αβ)
ϑ(ψ(ks , ls )as bs )
a1 ,...,at ∈k b1 ,...,bt ∈k 1≤s≤t
where β = 0 if (i, j) ∈
/ D(σ), and β = ψ(i, j) if (i, j) ∈ D(σ). Since for all a1 , . . . , at ∈ k
the mapping
Y
(b1 , . . . , bt ) 7→
ϑ(ψ(ks , ls )as bs )
1≤s≤t
38
CARLOS A. M. ANDRÉ
defines a linear character of the additive group of kt which is trivial if and only if
a1 = . . . = at = 0, we have
(
X Y
|kt | = q t , if a1 = . . . = at = 0,
ϑ(ψ(ks , ls )as bs ) =
0,
otherwise,
b ,...,b ∈k 1≤s≤t
1
t
and thus
χ{ij},α (gσ,ψ ) =
X
X Y
q j−i−1
q j−i−1
ϑ(αβ)
ϑ(ψ(k
,
l
)a
b
)
=
ϑ(αβ).
s s s s
t
q 2t
q
a ,...,a ∈k b ,...,b ∈k 1≤s≤t
t
1
1
t
This completes the proof of the following result.
Lemma 3.7. If (i, j) ∈ [[n]], α ∈ k× and (σ, ψ) ∈ Sn (k), then χ{ij},α (gσ,ψ ) = 0 unless
(i, k), (k, j) ∈
/ D(σ) for all i < k < j, in which case
( qj−i−1
if (i, j) ∈ D(σ),
nsti,j (σ) ϑ(αψ(i, j)),
χ{ij},α (gσ,ψ ) = qqj−i−1
otherwise.
nsti,j (σ) ,
q
In the general case, in virtue of Theorem 3.6 we deduce the following formulae for
the supercharacter values; here, we use the following terminology. For every π ⊢ [n],
we say that (k, l) ∈ [[n]] is a π-singular arc if (k, l) is covered by some arc (i, j) ∈ D(π)
and these two arcs share a common vertex (hence, either k = i, or l = j); otherwise,
we say that (k, l) is a π-regular arc. We denote by Sing(π) the subset of [[n]] consisting
of all π-singular arcs, and by Reg(π) the subset of [[n]] consisting of all π-regular arcs;
in particular, we have D(π) ⊆ Reg(π).
Theorem 3.8. If (π, φ), (σ, ψ) ∈ Sn (k), then χπ,φ (gσ,ψ ) = 0 unless D(σ) ⊆ Reg(π), in
which case
Y
χπ,φ (1)
χπ,φ (gσ,ψ ) = nstπ (σ)
ϑ φ(i, j)ψ(i, j) ;
q
(i,j)∈D(π)∩D(σ)
P
recall that nstπ (σ) = (i,j)∈D(π) nsti,j (σ).
We observe that, in the notation of the theorem, χπ,φ (1) = q m where
X
X
m=
(j − i − 1) =
(dim(i, j) − 1) = dim(π) − d(π)
(i,j)∈D(π)
(i,j)∈D(π)
where d(π) = |D(π)| is the number of arcs of π. Thus, the formula above can be
rewritten as
Y
q dim(π)−d(π)
(3j)
χπ,φ (gσ,ψ ) =
ϑ
φ(i,
j)ψ(i,
j)
.
q nstπ (σ)
(i,j)∈D(π)∩D(σ)
We note that nsti,j (σ) ≤ j − i + 1 = dim(i, j) − 1 for all (i, j) ∈ [[n]], and thus
nstπ (σ) ≤ dim(π) − d(π)
for all π, σ ⊢ [n]. On the other hand, since ϑ is a linear character of k+ , we have
|ϑ(α)| = 1 for all α ∈ k, and so up to the product by a root of unity the supercharacter
values are a nonnegative integers.
SUPERCHARACTERS AND SET PARTITION COMBINATORICS
39
3.6. A uncoloured supercharacter theory. As we mentioned in the Introduction,
most finite groups have more than one possible supercharacter theory, so there is not
really a canonical choice. For the unitriangular groups Un (k), there is a natural coarsening of the classical supercharacter theory (as we have just defined) that is integer
valued, and thus its use may be combinatorially more natural and advantageous. In
this supercharacter theory, supercharacters and superclasses of Un (k) are parametrised
by set partitions of [n]; for every π ⊢ [n], we define
X
[
1
π,φ
(3k)
χπ =
χ
and
K
=
Kπ,φ
π
(q − 1)d(π)
φ∈Colk (π)
φ∈Colk (π)
where q = |k| and where we denote by Colk (π) the set consisting of all k× -colourings
of π. Notice that Kπ contains the element gπ = 1 + eπ where we set
X
(3l)
eπ =
ei,j ∈ un .
(i,j)∈D(π)
Theorem 3.9. If X = Kπ : π ⊢ [n] and Y = χπ : π ⊢ [n] , then the pair (X , Y) is
a supercharacter theory of Un .
Proof. It is clear that X is partition of Un , and that Y an orthogonal set of characters
of Un ; moreover, we obviously have |X | = |Y|. If χ ∈ irr(Un ), then χ is a constituent
of χπ,φ for some (π, φ) ∈ Sn (k), and hence it is a constituent of χπ . Thus, it remains to
prove that for all π, σ ⊢ [n] the character χπ takes a constant value on Kσ . To see this,
we note that χπ is a superclass function with respect to the classical supercharacter
theory, and thus χπ takes a constant value on each superclass Kσ,ψ for ψ ∈ Colk (σ).
Therefore, it is enough to prove that χπ (gσ,ψ ) = χπ (gσ,ψ′ ) for all ψ, ψ ′ ∈ Colk (σ); by
Theorem 3.8 this is clear whenever D(σ) 6⊆ Reg(π). On the other hand, suppose that
D(σ) ⊆ Reg(π), and let ψ ∈ Colk (σ) be arbitrary. Then, by (3j) we know that
χπ (gσ,ψ ) =
1
(q − 1)d(π)
X
χπ,ϕ (gσ,ψ ) =
φ∈Colk (π)
where
ϑ(φ, ψ) =
Y
(i,j)∈D(π)∩D(σ)
q dim(π)−d(π)
(q − 1)d(π) q nstπ (σ)
X
ϑ(φ, ψ)
φ∈Colk (π)
ϑ φ(i, j)ψ(i, j) .
If D(π) ∩ D(σ) = {(i1 , j1 ), . . . , (it , jt )}, then we can identify any k× -colouring of D(π) ∩
D(σ) with a sequence (α1 , . . . , αt ) ∈ (k× )t , and thus we deduce that
Y
X
X
ϑ(αr ψ(ir , jr ))
ϑ(φ, ψ) = (q − 1)|D(π)|−|D(π)∩D(σ)|
α1 ,...,αt ∈k× 1≤r≤t
φ∈Colk (π)
= (q − 1)|D(π)|−|D(π)∩D(σ)|
X
α1 ,...,αt
= (q − 1)
|D(π)|−|D(π)∩D(σ)|
X
∈k×
α∈k×
Since ϑ is a nontrivial linear character of k+ , we have
1X
0 = ϑ, 1k =
ϑ(α),
q α∈k
Y
1≤r≤t
t
ϑ(α) .
ϑ(αr )
40
CARLOS A. M. ANDRÉ
and thus
X
ϑ(α) = −ϑ(1) = −1.
α∈k×
Since t = |D(π) ∩ D(σ)|, we conclude that
X
ϑ(φ, ψ) = (−1)|D(π)∩D(σ)| (q − 1)|D(π)|−|D(π)∩D(σ)| ,
φ∈Colk (π)
and so
(−1)d(π,σ) q dim(π)−d(π)
q nstπ (σ) (q − 1)d(π,σ)
where we write d(π, σ) = |D(π)∩D(σ)|. It follows that the value χπ (gσ,ψ ) is independent
of the colouring ψ ∈ Colk (σ), and this completes the proof.
χπ (gσ,ψ ) =
To illustrate, the supercharacter table of U4 (with respect to this uncoloured supercharacter theory) is as follows.
1/2/3/4 12/3/4 1/23/4 1/2/34 123/4 12/34 1/234 1234 13/2/4 1/24/3 124/3 134/2 13/24 14/2/3 14/23
1/2/3/4
12/3/4
1
1
1
−t−1
1
1
1
1
1
1
−t−1 −t−1
1/23/4
1
1
−t−1
1
−t−1
1
−t−1
1
1
−t−1
1/2/34
123/4
1
1
1
−t−1
1
−t−1
12/34
1
−t−1
1
−t−1
−t−1
−t−1
−t−1
1
−t−1 −t−1
1
1
1
−t−1
1
1
1
1
1
1
1
−t−1
1
1
1
1
1
1
−t−1 −t−1
1
1
1
1
1
1
−t−1
1
1
1
1
−t−1
1
1
1
1
1
1
1
−t−1
1
1
−t−1 −t−1
1
1
1
1
1
1
1
1
−t−1
−1
−t
−t−1
1
1
1
1
−t−1
−t−1
q
q
0
0
−t−1
−t−1
−t−1 −t−1 t−2
t−2
−t−1 t−2
t−2
t−2
1/234
1234
1
1
1
−t−1
−t−1
−t−1
t−2
t−2
13/2/4
1/24/3
q
q
0
q
0
0
q
0
0
0
0
0
0
0
0
0
124/3
q
−qt−1
0
0
0
0
0
0
q
−qt−1
134/2
13/24
q
q2
0
0
0
0
−qt−1
14/2/3
q2
0
14/23
q2
0
t−2 −t−3
−t−1
−qt−1
q
0
−qt−1 −qt−1
−1
−1
q
−qt
−qt
0
−qt−1
−qt−1
qt−2
0
−qt−1
q
0
0
0
qt−2
q
q2
0
0
0
0
0
0
0
0
0
0
0
−q 2 t−1
q
−q 2 t−1
0
−qt−1
q 2 t−2
q
0
0
0
0
0
0
0
0
0
0
−q 2 t−2 −qt−2
−qt−1
0
0
0
0
0
0
0
0
0
0
−q 2 t−1 qt−2
where q = |k| and t = q − 1.
3.7. Non-crossing and non-nesting set partitions. We know from Corollary 2.9
that
π,φ π,φ χ ,χ
= |Un ϑπ,φ ∩ ϑπ,φ Un |
for all (π, φ) ∈ Sn (k). Similarly to Lemma 3.5, the right G-orbits on u◦n can be described
as follows.
Lemma 3.10. If (i, j) ∈ [[n]] and α ∈ k× , then
ϑ{ij},α Un = ϑa : a ∈ αei,j + Ui,j
where Ui,j is the vector subspace
P of un spanned by {ei+1,j , . . . , ej−1,j }. More generally,
if (π, φ) ∈ Sn (k) and Uπ = (i,j)∈D(π) Ui,j , then
ϑπ,φ Un = ϑa : a ∈ eπ,φ + Uπ .
It follows (from this and from Lemma 3.5) that χπ,φ , χπ,φ equals q crs(π) where q = |k|
and crs(π) is the number of crossings of π. The following result is now immediate; in a
certain sense, it says that crossing number crs(π) measures how close the supercharacter
χπ,φ is to being irreducible.
SUPERCHARACTERS AND SET PARTITION COMBINATORICS
41
Theorem 3.11. If (π, φ) ∈ Sn (k), then χπ,φ , χπ,φ = q crs(π) . In particular, χπ,φ is an
irreducible character if and only if π is a non-crossing set partition.
An important example occurs when (π, φ) ∈ Sn (k) is the “maximal” non-crossing set
partition with standard representation:
α1
α2
αi
b
b
1
2
...
b
i
...
b
...
n−i+1
b
b
8
9
In this situation the supercharacter χπ,φ is irreducible and has maximal degree. In fact,
we have the following.
Theorem 3.12. Let (π, φ) ∈ Sn (k) be such that D(π) = (i, j) ∈ [[n]] : i + j = n + 1 .
Then, the supercharacter χπ,φ is irreducible and has degree χπ,φ (1) = q n(n−2)/4 if n is
2
even, or χπ,φ (1) = q (n−1) /4 if n is odd. Moreover, if χ is an irreducible character of
Un , then χ(1) ≤ χπ,φ (1) with equality if and only if χ = χσ,ψ for some (σ, ψ) ∈ Sn (k)
with D(σ) = D(π) or D(σ) = D(π) \ {(n/2, (n/2) + 1)} if n is even.
Proof. The first assertion is an immediate consequence of the previous theorem. On
the other hand, if χ is an irreducible character of Un , then χ is a constituent of χσ,ψ
for a unique (σ, ψ) ∈ Sn (k), and hence χ(1) ≤ χσ,ψ (1). By Theorem 3.6,
Y
χσ,ψ (1) =
q j−i−1
(i,j)∈D(σ)
which is certainly not greater that χπ,φ (1); further, it is easily seen that the equality
occurs if and only if D(σ) is as indicated.
As for the case of non-crossing set partitions, also the non-nesting set partitions
of [n] can be characterised in terms of the supercharacter theory of Un . In fact, if
(π, σ) ∈ Sn (k), then it is well-known that |χπ,φ (g)| ≤ χπ,φ (1) for all g ∈ Un (see [26,
Lemma 2.15]); the set
Z(χπ,φ ) = g ∈ Un : |χπ,φ (g)| = χπ,φ (1)
is a subgroup of Un and can be described as the set of all g ∈ Un such that g · ς ∈ Cς
for all ς ∈ Lπ,φ ; recall that Lπ,φ = CUn · επ,φ is a CUn -module affording χπ,φ . In light
of the fact that L(ϑπ,φ ) = StabUn (Cεπ,φ ), it is an easy exercise to show that
\
\
L(gϑπ,φ ).
gL(ϑπ,φ )g −1 =
Z(χπ,φ ) =
g∈Un
g∈Un
On the other hand, Theorem 3.8 implies that
[
Z(ϑπ,φ ) =
Kσ,ψ ;
(σ,φ)∈Sn (k)
nstπ (σ)=0
indeed, if (σ, ψ) ∈ Sn (q) and g ∈ Kσ,ψ then |χπ,φ (g)| = χπ,φ (1) if and only if nstπ (σ) = 0.
In particular, we deduce that Kπ,φ ⊆ Z(χπ,φ ) if and only if nstπ (π) = 0; in other words,
we have the following.
42
CARLOS A. M. ANDRÉ
Theorem 3.13. If (π, φ) ∈ Sn (k), then Kπ,φ ⊆ Z(χπ,φ ) if and only if π is a non-nesting
set partition.
Exercises.
3.1. Let (i, j) ∈ [[n]], and let α ∈ k× . Show that the superclass K{ij},α of Un is in
fact the conjugacy class of xi,j (α) = 1 + αei,j ∈ Un . Also, show that the elementary
character χ{ij},α is irreducible.
3.2. ([8, Section 2]) Let π ⊢ [n], and define R(π) = [[n]] \ S(π) where
[
S(π) =
{(i, j + 1), . . . , (i, n), (1, j), . . . , (i − 1, j)}.
(i,j)∈D(π)
For every (i, j) ∈ R(π), and every a ∈ un , define ∆πi,j (a) to be the determinant
ai,j
·
·
·
a
a
i,j
i,j
σ(t)
σ(1)
ai ,j
·
·
·
a
a
i
,j
i
,j
1
1
1
σ(1)
σ(t)
∆πi,j (a) = ..
..
.. .
.
. ait ,jσ(1) · · · ait ,jσ(t) ait ,j where {(i1 , j1 ), . . . , (it , jt )} = (k, l) ∈ D(π) : i < k < l < j , i1 < . . . < it , and σ ∈ St
is the unique permutation such that jσ(1) < . . . < jσ(t) . Moreover, let φ : π → k× is a
k× -colouring of π. Prove that
Y

(−1)t sgn(σ)φ(i, j)
φ(ir , jr ), if (i, j) ∈ D(π),
π
∆i,j (eπ,φ ) =
1≤r≤t

0,
otherwise,
for all (i, j) ∈ R(π), and conclude that
Kπ,φ = g ∈ Un : ∆πi,j (g − 1) = ∆πi,j (eπ,φ ) for all (i, j) ∈ R(π) .
3.3. [7, Theorem 1] Let π ⊢ [n], and recall that Reg(π) = [[n]] \ Sing(π) where
[
Sing(π) =
{(i, i + 1), . . . , (i, j − 1), (i + 1, j), . . . , (j − 1, j)}.
(i,j)∈D(π)
For every (i, j) ∈ Reg(π), and every a ∈ un , define ∇πi,j (a) to be the determinant
ai1 ,j ai1 ,j
·
·
·
a
i
,j
1
σ(1)
σ(t)
..
..
.. .
. ∇πi,j (a) = .
ait ,j ait ,jσ(1) · · · ait ,jσ(t) ai,j ai,jσ(1) · · · ai,jσ(t) where {(i1 , j1 ), . . . , (it , jt )} = (k, l) ∈ D(π) : k < i < j < l , i1 < . . . < it , and σ ∈ St
is the unique permutation such that jσ(1) < . . . < jσ(t) . Moreover, let φ : π → k× is a
k× -colouring of π. Prove that
Y

(−1)t sgn(σ)φ(i, j)
φ(ir , jr ), if (i, j) ∈ D(π),
∇πi,j (eπ,φ ) =
1≤r≤t

0,
otherwise,
for all (i, j) ∈ Reg(π), and conclude that
Un ϑπ,σ Un = ϑa : a ∈ un , ∇πi,j (a) = ∇πi,j (eπ,φ ) for all (i, j) ∈ Reg(π) .
SUPERCHARACTERS AND SET PARTITION COMBINATORICS
43
3.4. Let G = 1 + A be an algebra group over k, let ϑ ∈ A◦ , and let
O(ϑ) = g −1ϑg : g ∈ G
be the conjugation orbit of ϑ. Define an alternating k-bilinear form Bϑ : A × A → k
by Bϑ (a, b) = ϑ(ab − ba) for all a, b ∈ A, and let
C(ϑ) = a ∈ A : ϑ(ab) = ϑ(ba) for all b ∈ A
be the radical of this form.
(1) Prove that C(ϑ) is a subalgebra of A, and that C(ϑ) = 1+C(ϑ) is the centraliser
of ϑ with respect to the conjugation action. Conclude that |O(ϑ)| is a power of
q = |k| with even exponent.
(2) Define the function κϑ : G → C by
X
1
κϑ (g) = p
ϑ̂(g)
|O(ϑ)| ϑ′ ∈O(ϑ)
for all g ∈ G. Prove that κϑ : ϑ ∈ A◦ is an orthonormal basis of complex
vector space cf(G) consisting of all class functions of G. 6
3.5. [10, Section 7] Let π ⊢ [n], let φ : D(π) → k× be a k× -colouring of π, and let
ϑ = ϑπ,φ ∈ u◦n . For any t ∈ N, we define a t-crossing of a set partition π ⊢ [n] to be
a sequence i0 < i1 < i2 < i3 < · · · < it < it+1 < it+2 where (ir , ir+2 ) ∈ D(π) for all
0 ≤ r ≤ t; we refer to t as the length of the t-crossing. A t-crossing of π is called a
maximal crossing if it is cannot be extended to a (t + 1)-crossing. Let C(π) be the
subset of [[n]] consisting of all (i, j) ∈ [[n]] for which there exists a maximal crossing
i0 < i1 < i2 < i3 < · · · < it < it+1 < it+2 of π such that (i, j) = (ir , ir+1 ) for some
1 ≤ r ≤ t with t−r odd; that is, the (t−r)-crossing ir < ir+1 < · · · < it < it+1 < it+2 has
odd length. Moreover, define D ′ (π) to be the subset of [[n]] consisting of all (i, j) ∈ [[n]]
for which there exists a maximal crossing i0 < i1 < i2 < i3 < · · · < i2s−1 < i2s < i2s+1
of π with odd length and such
that i = i0 and
j = i1 . Finally, let V be the vector
subspace of un spanned by ei,j : (i, j) ∈ C(π) , and define H(π) = 1 + (V + L(π))
where we write L(π) = L(ϑ). Prove that the following hold.
(1) H(π) is an algebra subgroup of Un , and ϑ̂ : H(π) → C is a linear character of
H(π).
(2) If κπ,φ denotes the Kirillov function of Un associated with ϑ, then κπ,φ =
n
(ϑ̂), and hence κπ,φ is an irreducible character of Un .
IndUH(π)
(3) κπ,φ is an irreducible constituent of the supercharacter χπ,φ with multiplicity
′
κπ,φ , χπ,φ = q (1/2)(crs(π)−|D (π)|) .
6These functions κ are known as the Kirillov functions of the algebra group G. It was conjectured
ϑ
by Kirillov [32] that, in the case where G = Un the Kirillov functions are exactly the irreducible
characters of G. However, it is now known that this cannot be correct in general; for example, all
Kirillov functions of Un (F2 ) have real values, but it is known that U13 (F2 ) has a non-real irreducible
character (see [28] and [29]). Nevertheless, many of the Kirillov functions for the groups Un are
actually irreducible characters; on the other hand, it is true that all irreducible characters of any
algebra group G are given by a slightly variation of the Kirillov functions provided that Ap = 0 where
p is the characteristic of k (see [7] or [38]).
44
CARLOS A. M. ANDRÉ
(4) The
(a)
(b)
(c)
following are equivalent:
The supercharacter χπ,φ has a unique irreducible constituent.
All the maximal crossings of π have even length.
The set D ′ (π) is empty.
4. The Hopf algebra of superclass functions
As seen in the proof of Theorem 3.9, for every n ∈ N and every π ⊢ [n], the
(uncoloured) supercharacter χπ of Un is a Q-valued function χπ : Un → Q; in fact, if
π ′ ⊢ [n], then χπ takes the constant value
′
(4a)
χπ (gπ′ ) =
(−1)d(π,π ) q dim(π)−d(π)
q nstπ (π′ ) (q − 1)d(π,π′ )
on the superclass Kσ . We define SCn = SCn (k) to be the Q-vector space spanned by
the (uncoloured) supercharacters of the unitriangular
group Un ; by the orthogonality
π
of supercharacters we see that χ : π ⊢ [n] is a Q-basis of SCn . Furthermore, it
is also clear that SCn consists of all Q-valued superclass functions of Un ; in other
words, we have SCn = scfQ (Un ). In fact, it follows from the general properties of
supercharacter theories that SCn also has a distinguished Q-basis consisting of all the
superclass characteristic functions. For every π ⊢ [n], we denote by κπ the characteristic
function of the superclass Kπ , and recall that for all g ∈ Un the value κπ (g) is given by
(
1, if g ∈ Kπ ,
κπ (g) =
0, if g ∈
/ Kπ ;
in fact, κπ are uniquely
determined
by the values κπ (gπ′ ) = δπ,π′ for all π ′ ⊢ [n]. It
is obvious that κπ : π ⊢ [n] is a Q-basis of SCn , and this implies that SCn is indeed
equal to scfQ (Un ).
We extend the definition of SCn to all nonnegative integers by setting SC0 = Q; we
agree that ∅ is the unique set partition of [0] = ∅, and set χ∅ = κ∅ = 1 ∈ Q. Then, we
define the (graded) C-vector space
M
SCn ,
SC =
n∈N0
and we refer to SC as the Q-vector space of superclass functions. Our aim is to introduce
a structure of (combinatorial) Hopf Q-algebra on SC which as we shall see turns out
to be isomorphic to the Hopf Q-algebra NCSym(X) introduced by Rosas and Sagan in
[36]. We start by describing some of the basic theory of Hopf algebras.
4.1. Hopf algebra basics. Hopf algebras arise naturally in combinatorics and algebra, where there are “things” that break into parts which can also be put together with
some compatibility between operations [30]. They have emerged as a central object of
study in algebra through quantum groups [17, 19, 39] and in combinatorics [2, 4, 25].
Hopf algebras find applications in diverse fields such as algebraic topology, representation theory, and mathematical physics. We suggest the first few chapters of [39] for
a motivated introduction and [35] as a basic text; each has extensive references. The
present subsection gives definitions to make our exposition self-contained.
Let k be an arbitrary field. A k-coalgebra is a k-vector space C with two k-linear
maps: the coproduct ∆ : C → C ⊗ C and the coidentity ε : C → k. The coproduct must
SUPERCHARACTERS AND SET PARTITION COMBINATORICS
45
be coassociative, in the sense that the diagram
C
∆
C⊗C
/
∆⊗id
∆
C⊗C
/
id ⊗∆
C⊗C
is commutative, that is,
(∆ ⊗ id) ◦ ∆ = (id ⊗∆) ◦ ∆;
equivalently, for every c ∈ C we have
X
X
X
∆(c′i ) ⊗ c′′i =
c′i ⊗ ∆(c′′i )
where ∆(c) =
c′i ⊗ c′′i .
i
i
i
On the other hand, the coidentity must be compatible with the coproduct, that is, the
diagram
∆
/C⊗C
C LLL
LL
LL
ε⊗id
LL
∆
LL id
L%
C ⊗ C id ⊗ε / C
is commutative. Thus, we must have
(ε ⊗ id) ◦ ∆ = (id ⊗ε) ◦ ∆ = id
where we naturally identify C with k ⊗ C and C ⊗ k; more explicitly, for every c ∈ C we
have
X
X
X
c=
ε(c′i )c′′i =
ε(c′′i )c′i
where ∆(c) =
c′i ⊗ c′′i .
i
i
i
A map φ : C → D between k-coalgebras is a homomorphism of k-coalgebras if φ is
k-linear and
∆D ◦ φ = (φ ⊗ φ) ◦ ∆C
where ∆C and ∆D are the coproducts of C and D, respectively; thus, a k-linear map
φ : C → D is a homomorphism of k-coalgebras if the diagram
C
∆C
/
C ⊗C
φ
φ⊗φ
D
/
∆D
D⊗D
is commutative.
We say that an associative k-algebra A is a k-bialgebra if A is also a k-coalgebra
and the operations are compatible:
• The coproduct ∆ : A → A ⊗ A is a homomorphism of k-algebras where we
define (a ⊗ b)(c ⊗ d) = ac ⊗ bd for all a, b, c, d ∈ A.
• The coidentity ε : A → k is a homomorphism of k-algebras.
• We have
∆ ◦ u = (u ⊗ u) ◦ δ and ε ◦ u = id
where u : k → A is defined by u(α) = α · 1A for all α ∈ k, and δ : k → k ⊗ k is
defined by δ(α) = α ⊗ α for all r ∈ k.
For example,
46
CARLOS A. M. ANDRÉ
(1) the group algebra kG of a finite group G becomes a k-bialgebra under the maps
defined by
∆(g) = g ⊗ g and ε(g) = 1
for all g ∈ G;
(2) the polynomial algebra k[x1 , . . . , xn ] on commuting indeterminates x1 , . . . , xn
over k becomes a k-bialgebra under the operations
∆(xi ) = xi ⊗ 1 + 1 ⊗ xi
and ε(xi ) = 0
for all 1 ≤ i ≤ n.
A k-bialgebra A is said to be graded if there is a decomposition
M
A=
An
n≥0
as a direct sum of vector subspaces satisfying
Am An ⊆ Am+n ,
u(k) ⊆ A0 ,
∆(An ) ⊆
M
Ai ⊗ An−i
and ε(An ) = 0
0≤i≤n
for all m, n ≥ 1; A is said to be connected if A0 = u(k). By way of example, the
polynomial algebra k[x1 , . . . , xn ] is graded according to polynomial degree.
If A is a k-bialgebra, then an antipode of A is a k-linear map η : A → A such that
for all a ∈ A
X
X
X
(4b)
η(a′i )a′′i =
a′′i η(a′i ) = ε(a) · 1A
where ∆(a) =
a′i ⊗ a′′i .
i
i
i
A k-bialgebra with an antipode is called a Hopf k-algebra. For example, the group bialgebra kG has antipode given by η(g) = g −1 , and the polynomial bialgebra k[x1 , . . . , xn ]
has antipode given by η(xi ) = −xi .
If A is a connected graded k-bialgebra, then (4b) can be solved inductively to give
η(α · 1A ) = α · 1 for all α ∈ k, and
X
η(a) = −a +
η(ai )a′n−i
1≤i<n
where a ∈ An is arbitrary and such that
∆(a) = a ⊗ 1 + 1 ⊗ a +
X
ai ⊗ an−i .
1≤i<n
Thus, every connected graded k-bialgebra has an antipode, and is automatically a Hopf
k-algebra.
4.2. Symmetric functions in non-commuting variables. Consider the Q-algebra
QhhXii consisting of formal power series in a set X = {x1 , x2 , x3 , . . .} of countably
many non-commuting variables over the rational field Q. The subalgebra of symmetric
functions in the non-commuting variables X consists of all elements invariant under
permutation of the variables and of bounded degree. More precisely, for every n ∈ N,
there is an action of the symmetric group Sn on the Q-algebra QhhXii given by
σ · f (x1 , x2 , . . .) = f (xσ(1) , xσ(2) , . . .)
where σ(i) = i for i > n. We say that f is symmetric if it is invariant under the
action of Sn for all n ≥ 1; in other words, we let S∞ denote the infinite symmetric
group consisting of all bijections σ : N → N with finitely many non-fixed points, and
let S∞ act naturally by automorphisms of the Q-algebra QhhXii. Then, we define
SUPERCHARACTERS AND SET PARTITION COMBINATORICS
47
the Q-algebra of symmetric functions in non-commuting variables NCSym = NCSym(X)
to be the subalgebra consisting of all elements invariant under the S∞ -action and of
bounded degree. These symmetric functions have a theory analogous to the ordinary
theory of symmetric functions, and were first studied by Wolf [46] in 1936. Her goal
was to provide an analogue of the fundamental theorem of symmetric functions in
this context. The concept then lay dormant for over 30 years until Bergman and
Cohn generalised Wolf’s result [15]. Still later, Kharchenko [31] proved that if V is a
positively graded vector space and G is a group of gradation-preserving automorphisms
of the tensor algebra T V , then the algebra of invariants of G is also a tensor algebra.
Anick [11] then showed that one could remove the hypothesis about G preserving the
grading. Most recently, Gebhard and Sagan [23] revived these ideas as a tool for
studying Stanley’s chromatic symmetric function [40, 42].
If π ⊢ [n], then a monomial of shape π in QhhXii is a product xi1 xi2 · · · xin where
ir = is if and only if r and s are in the same block in π (that is if and only if
(r, s) ∈ D(π)). For example, x1 x2 x1 x2 is a monomial of shape 13/24 ⊢ [4]. We define
the monomial symmetric function mπ = mπ (X) to be the sum of all monomials of shape
π. For example,
m13/24 = x1 x2 x1 x2 + x2 x1 x2 x1 + x1 x3 x1 x3
+ x3 x1 x3 x1 + x2 x3 x2 x3 + x3 x2 x3 x2 + · · · .
Note that these functions are exactly those gotten by symmetrizing a monomial, and
so are invariant under the action of Sn defined previously. It follows easily that they
form a basis for the Q-algebra NCSym.
Remark 4.1. Other basis for NCSym may be defined as non-commuting versions of the
analogous well-known basis in the ordinary theory of symmetric functions: power-sum,
elementary, and others. For the first of these, for a given set partition π ⊢ [n], we
define the power-sum function pπ to be the sum of all monomials xi1 xi2 · · · xin where
ir = is if (r, s) ∈ D(π). To illustrate,
p13/24 = x1 x2 x1 x2 + x2 x1 x2 x1 + x41 + x42 + · · · = m13/24 + m1234 .
It is not clear why these functions form a basis for NCSym; for the proof, as well as for
the following analogues, we refer to the Rosas and Sagan paper [36].
On the other hand, we define the elementary symmetric functions eπ to be the sum
of all monomials xi1 xi2 · · · xin where ir 6= is if (r, s) ∈ D(π). By way of example,
e13/24 = x1 x1 x2 x2 + x2 x2 x1 x1 + x1 x2 x2 x1 + x2 x1 x2 x1 + · · ·
= m12/34 + m14/23 + m12/3/4 + m14/2/3 + m1/23/4 + m1/2/34 + m1/2/3/4 .
It is possible to exhibit a relation of these symmetric functions to the corresponding
ordinary symmetric functions; for a detailed discussion see [36]. This is done via
the forgetful or projection map ρ : QhhXii → Q[[X]] which merely lets the variables
commute; here, Q[[X]] denotes the usual Q-algebra consisting of formal power series on
commuting variables x1 , x2 , . . . In fact, up to well-determined positive integer constant
(which depends only on the sizes of the blocks of π), the images ρ(mπ ), ρ(pπ ) and
ρ(eπ ) are precisely the corresponding monomial, power-sum and elementary symmetric
functions associated to the integer partition λ = λ(π) ⊢ n given by the ordered sizes
of the blocks of π.
48
CARLOS A. M. ANDRÉ
The Q-algebra NCSym has a natural grading
M
NCSym =
NCSymn
n≥0
where NCSymn is the Q-vector subspace spanned by mπ : π ⊢ [n] ; recall that this
means that NCSym0 = Q · 1 and NCSymm · NCSymn ⊆ NCSymm+n for all m, n ∈ N0 . As an
example, we calculate
m13/2 m124/3 = m13/2/457/6 + m13457/2/6 + m13/2457/6 + m136/457/2
+ m13/26/457 + m13457/26 + m136/2457 .
In the general situation, we have the proposition below where we use the following notation. If π = B1 /B2 / . . . /Bℓ and π ′ = B1′ /B2′ / . . . /Bℓ′ ′ are set partitions of nonnegative
integers m and n respectively, then π/π ′ ⊢ [m + n] is defined by
π/π ′ = B1 /B2 / . . . /Bℓ /(B1′ + m)/(B2′ + m)/ . . . /(Bℓ′ + m)
where I + m = i + m : i ∈ I for all I ⊆ [n]. For example, if π ⊢ [6] and π ′ ⊢ [8] have
representations
(4c)
and
b
b
b
b
b
b
1
2
3
4
5
6
b
b
b
b
b
b
b
b
1
2
3
4
5
6
7
8
respectively, then π/π ′ ⊢ [14] is represented as
b
b
b
b
b
b
b
b
b
b
b
b
b
b
1
2
3
4
5
6
7
8
9
10
11
12
13
14
On the other hand, for every n ∈ N0 and every π, π ′ ⊢ [n], we define π ≤ π ′ if each block
B of π is contained in some block B ′ of π ′ ; then, with this order the set π : π ⊢ [n]
forms a ranked poset with rank function given by π 7→ n−ℓ(π) where ℓ(π) is the length
of π. The meet (greatest lower bound) and join (least upper bound) operations are
denoted ∨ and ∧, respectively. For example, if π = 138/24/5/67 and π ′ = 1/238/4567,
then π and π ′ are not comparable in the partial order on set partitions. We calculate
that
π ∨ π ′ = 1/2/38/4/5/67 and π ∧ π ′ = 12345678.
Notice that,
B1 /B2 / . . . /Bℓ and π ′= B1′ /B2′ / . . . /Bℓ′ ′ , then the parts of π ∧π ′ are
if π =
′
given by Bi ∩ Bj : 1 ≤ i ≤ ℓ, 1 ≤ j ≤ ℓ′ . Furthermore, we note that 0̂n = 1/2/ . . . /n
is the unique minimal element of this poset, whereas 1̂n = 12 . . . n is its unique maximal
element.
The proof of the following result is straightforward, and we leave it as an exercise
for the reader.
Proposition 4.2. If π ⊢ [m] and π ′ ⊢ [n] for any m, n ∈ N, then
X
mσ .
(4d)
mπ mπ′ =
σ⊢[m+n]
σ∧(1̂m /1̂n )=π/π ′
SUPERCHARACTERS AND SET PARTITION COMBINATORICS
49
It follows from (4d) that NCSym is indeed a Q-subalgebra of QhhXii. Our point
of departure is to consider NCSym as a Hopf algebra so that we may examine it from
another perspective. Besides the associative product on NCSym, it is also possible to
define a coproduct map ∆ : NCSym → NCSym ⊗ NCSym as follows.
Firstly, let Y = {y1 , y2, y3 , . . .} and Z = {z1 , z2 , z3 , . . .} be two disjoint countable
infinite sets of non-commuting variables, and assume that we have relations yi zj = zj yi
for all i, j ∈ N; formally, we consider the quotient Q-algebra A(Y, Z) = QhhY, Zii/I
where I is the ideal of QhhY, Zii generated by the elements yi zj − zj yi . It is clear that
every f (Y, Z) ∈ A(Y, Z) can be written as a sum
X
(4e)
f (Y, Z) =
fi′ (Y )fi′′ (Z)
i
fi′ (Y
fi′′ (Z)
where
) ∈ QhhY ii and
∈ QhhZii.
Now, for every f (X) ∈ QhhXii, we can define the element
f (Y, Z) = f (y1, z1 , y2 , z2 , . . .)
of AhhY, Zii, and obtain fi′ (Y ) ∈ QhhY ii and fi′′ (Z) ∈ QhhZii such that f (Y, Z)
decomposes as in (4e). Then, we consider fi′ (X), fi′′(X) ∈ QhhXii, and define the
coproduct ∆(f ) ∈ QhhXii ⊗ QhhXii by
X
(4f)
∆(f ) =
fi′ ⊗ fi′′ .
i
The advantage of defining the coproduct in this way is that ∆ is clearly coassociative
and a homomorphism of Q-algebras. The next proposition gives the formula for the
coproduct of a monomial symmetric function in QhhXii; the following notation is
needed. If π ⊢ [n] (for n ∈ N), π = B1 /B2 / . . . /Bℓ(π) , and I = {i1 , . . . , ik } is a subset
of [ℓ(π)], then we define the set partition
(4g)
πI = stJ (Bi1 )/ stJ (Bi2 )/ . . . / stJ (Bik ) ⊢ [m]
where J = Bi1 ∪ · · · ∪ Bik , m = |J| and stJ denotes the unique order preserving
map stJ : J → [m] (we refer to stJ as the straightening map of J). To illustrate, if
π = 1368/2/4/579 ⊢ [9], then
π{1,2} = 1345/2,
π{2,4} = π{3,4} = 1/234 and π{1,2,4} = 1357/2/468.
Proposition 4.3. If n ∈ N and π ⊢ [n], then
X
(4h)
∆(mπ ) =
mπI ⊗ mπI c
I⊆[ℓ(π)]
where I c = [ℓ(π)] \ I. In particular, we have
M
∆(NCSymn ) ⊆
NCSymi ⊗ NCSymn−i .
0≤i≤n
Proof. See [3, Section 6.2], or [13, Theorem 4.1].
By way of example, we have
∆(m14/2/3 ) = m14/2/3 ⊗ m∅ + 2 m13/2 ⊗ m1 + m12 ⊗ m1/2
+ m1/2 ⊗ m12 + 2 m1 ⊗ m13/2 + m∅ ⊗ m14/2/3 ;
notice that m∅ = 1 is the identity of the Q-algebra NCSym.
It is known ([3, Section 6.2], or [13, Theorem 4.1]) that NCSym, endowed with
the product and coproduct as above, is a Hopf algebra over Q where the coidentity
50
CARLOS A. M. ANDRÉ
ε : NCSym → Q is given by ε(m∅ ) = 1 and ε(mπ ) = 0 for all π ⊢ [n] for n ∈ N, and the
antipode is inherited from the grading.
In what follows, we consider the power-sum Q-basis pπ : π ⊢ [n], n ∈ N of NCSym;
recall that pπ for π ⊢ [n], n ∈ N, is defined to be the sum of all monomials xi1 xi2 · · · xin
where ir = is if (r, s) ∈ D(π). From the definition it follows easily that
X
pπ =
mσ
σ≥π
for all π ⊢ [n], n ∈ N, and so it is routine to check that the following formulae hold.
Proposition 4.4. If π ⊢ [m] and π ′ ⊢ [n] for m, n ∈ N0 , then
(4i)
pπ pπ′ = pπ/π′ ,
and
(4j)
X
∆(pπ ) =
pπI ⊗ pπI c .
I⊆[ℓ(π)]
4.3. Representation theoretic functors on SC. We will focus on a number of representation theoretic operations on the vector space SC which are crucial for the definition
of its Hopf algebra structure. If J is any subset of [n] (possibly J = ∅), we write UJ
to denote the subgroup of Un consisting of all matrices g ∈ Un satisfying the gi,j = 0
for all (i, j) ∈
/ J × J. For example, if J = {1, 2, 5, 6} ⊆ [6], then an element of UJ is
represented by the generic matrix


1 ∗ 0 0 ∗ ∗
0 1 0 0 ∗ ∗


0 0 1 0 0 0

.
0 0 0 1 0 0
0 0 0 0 1 ∗
0 0 0 0 0 1
It is obvious that UJ is an algebra subgroup of U; indeed, UJ = 1+uJ where uJ denotes
the k-subalgebra of Un consisting of all matrices a ∈ un satisfying the ai,j = 0 for all
(i, j) ∈
/ J × J. subalgebra. Notice also that U[n] = Un and u[n] = un , whereas U∅ = 1
and u∅ = 0.
If |J| = m, then the straightening map stJ : J → [m] induces a canonical isomorphism of groups stJ : UJ → Um , and also a k-linear isomorphism stJ : uJ → um , obtained by reordering the rows and columns; obviously, we have stJ (1 + a) = 1 + stJ (a)
for all a ∈ uJ . In the above example, we have stJ (UJ ) = U4 and


1 g1,2 0 0 g1,5 g1,6


1 g1,2 g1,5 g1,6
0 1 0 0 g2,5 g2,6 


0 1 g2,5 g2,6 
0 
stJ
0 0 1 0 0
.
 7−→ 

0 0
1 g5,6 
0 
0 0 0 1 0
0 0 0 0 1 g 
0 0
0
1
5,6
0
0
0 0
0
1
P
In particular, if π ⊢ [n] and D(π) ⊆ J × J, then eπ = (i,j)∈D(π) ei,j is an element of
uJ , and thus we obtain an element stJ (eπ ) ∈ um . Since stJ (ei,j) = estJ (i),stJ (j) for all
(i, j) ∈ J × J, we see that
X
stJ (eπ ) =
estJ (i),stJ (j) .
(i,j)∈D(π)
SUPERCHARACTERS AND SET PARTITION COMBINATORICS
51
In fact, since (stJ (i), stJ (j)) : (i, j) ∈ D(π) is a basic subset of [[m]], there is a
unique set partition σ ⊢ [m] such that stJ (eπ ) = eσ ; henceforth, we denote this set
partition by stJ (π). For instance, if J = {1, 2, 5, 6}, then stJ (125/3/4/6) = 123/4 ⊢ [4].
Conversely, if σ ⊢ [m] for some 0 ≤ m ≤ n and J ⊆ [n], then there exists a unique
set partition π ⊢ [n] with D(π) ⊆ J × J and such that stJ (π) = σ; as it is natural,
we write π = st−1
J (σ). Notice that, for every J ⊆ [n] with |J| = m, the set partition
−1
stJ (1̂m ) ⊢ [n] has blocks J and {k} for k ∈ [n] \ J, whereas st−1
J (0̂m ) = 0̂n .
′
On the other hand, let J and J be two disjoint subsets of [n]. Then, UJ ∩UJ ′ = 1, and
thus UJ UJ ′ is a subgroup of Un which is the (internal) direct product of the subgroups
UJ and UJ ′ . We denote this subgroup by UJ/J ′ , and note that UJ/J ′ is canonically
isomorphic to the (external) direct product U|J| × U|J ′ | via the map stJ/J ′ : UJ/J ′ →
U|J| × U|J ′ | defined by
stJ/J ′ (gh) = (stJ (g), stJ ′ (h))
for all g ∈ UJ and all h ∈ UJ ′ . For our purposes, the situation where J ′ = J c = [n] \ J
is of central interest. In this case, if |J| = m, then we obtain the group isomorphism
stJ/J c : UJ/J c → Um × Un−m . For example, if J = {1, 2, 5, 6}, then stJ/J c (UJ/J c ) =
U4 × U2 , and


1 g1,2 0 0 g1,5 g1,6



1 g1,2 g1,5 g1,6 0 1 0 0 g2,5 g2,6 


0 1 g2,5 g2,6  1 g3,4 
0  stJ/J c 
0 0 1 g3,4 0

.


,
 7−→ 

0 0
1 g5,6  0 1 
0
0 
0 0 0 1
0 0 0 0
0 0
0
1
1 g 
5,6
0
0
0
0
0
1
We observe that, the direct product
Um × Un−m can be embedded in a canonical way
in Un via the mapping (g, h) 7−→ g0 h0 ; henceforth, we identify Um × Un−m with its
image in Un under this homomorphism; hence,
g 0
Um × Un =
: g ∈ Um , h ∈ Un−m .
0 h
With this identification, the isomorphism stJ/J c : UJ/J c → Um × Un−m determines an
injective homomorphism of groups stJ/J c : UJ/J c → Un with image stJ/J c (UJ/J c ) =
Um × Un−m . To illustrate, in the example above,




1 g1,2 g1,5 g1,6 0 0
1 g1,2 0 0 g1,5 g1,6
0 1 g2,5 g2,6 0 0 
0 1 0 0 g2,5 g2,6 

 st c 

J/J
1 g5,6 0 0 
0 
0 0
0 0 1 g3,4 0
.
 7−→ 

0
1 0 0 
0
0 
0 0
0 0 0 1
0 0
0 0 0 0
0
0 1 g3,4 
1 g5,6 
0 0
0
0 0 1
0 0 0 0
0
1
The analogous definitions are valid for the k-algebra un : if J and J ′ are two disjoint
subsets of [n], then we define uJ/J ′ = uJ + uJ ′ ; this is a k-subalgebra of un , and there
is a canonical isomorphism of k-algebras stJ/J ′ : uJ/J ′ → u|J| × u|J ′| defined by
stJ/J ′ (a + b) = (stJ (a), stJ ′ (b))
for all a ∈ uJ and all b ∈ uJ ′ . Observe that UJ/J ′ = 1 + uJ/J ′ , and hence UJ/J ′ is an
algebra subgroup of Un . In the case where J ′ = J c , then we obtain an isomorphism of
k-algebras
stJ/J c : uJ/J c → um × un−m
52
CARLOS A. M. ANDRÉ
where m = |J|; as above, we naturally identify um × un−m with the k-subalgebra of un
such that Um × Un−m = 1 + (um × un−m ).
In particular, if σ ⊢ [m] and τ ⊢ [n − m] for 0 ≤ m ≤ n, then the set partition
σ/τ ⊢ [n] indexes the element
eσ 0
eσ/τ =
0 eτ
of um × un , and thus for every J ⊆ [n] with |J| = m we obtain an element st−1
J/J c (eσ/τ )
−1
of uJ/J c . In fact, by the definition we have stJ/J c (eσ/τ ) = est−1 (σ) + est−1c (τ ) , and thus
J
J
st−1
(e
)
=
e
where
π
⊢
[n]
is
uniquely
determined
by
σ,
τ
and
J.
In
fact, it is not
π
J/J c σ/τ
−1
−1
hard to prove that π = stJ (σ) ∨ stJ c (τ ); recall that ∨ denotes the join in the poset of
set partitions of [n]. For simplicity, we define
−1
−1
st−1
J/J c (σ/τ ) = stJ (σ) ∨ stJ c (τ );
−1
thus, with this notation we have st−1
. For example, if J =
J/J c (eσ/π ) = est
c (σ/τ )
J/J
{1, 2, 5, 6}, σ = 14/23 and τ = 12, then
st−1
J (σ) = 16/25/3/4,
st−1
J c (τ ) = 1/2/34/5/6 and
st−1
J/J c (σ/τ ) = 16/25/34.
Conversely, let J ⊆ [n], and let π ⊢ [n] be such that eπ ∈ uJ/J c . If m = |J|, then
stJ/J c (eπ ) ∈ um × un−m , and we have
eσ 0
stJ/J c (eπ ) =
= eσ/τ
0 eτ
where σ ⊢ [m] and τ ⊢ [n − m] are uniquely determined by π and J. Indeed, the
condition eπ ∈ uJ/J c is equivalent to saying that every block ofπ is either a subsetof J or
a subset of J c . Therefore, if π = B1 /B2 / . . . /Bℓ(π) and I = i ∈ [ℓ(π)] : Bi ⊆ J , then
σ = πI whereas τ = πI c (recall (4g) for the definitions). To illustrate, if J = {1, 2, 5, 6}
and π = 16/25/34 ⊢ [6], then ℓ(π) = 3 and I = {1, 2}, and thus we obtain
σ = πI = π{1,2} = 14/23 and τ = πI c = π{3} = 12;
notice that σ/τ = 14/23/56. In the general situation, as it is natural, we can also write
σ/τ = stJ/J c (π).
Next, we introduce the promised representation theoretic operations on the Q-vector
space
M
SCn .
SC =
n∈N0
We begin by recalling that for every m, n ∈ N0 we are identifying the direct product
Um × Un with a subgroup of Um+n ; moreover, since Um × Un is an algebra group over k,
it has a supercharacter theory in which every superclass function has the form φ × ψ
for φ ∈ SCm and ψ ∈ SCn . In fact, it is clear that the mapping φ ⊗ ψ 7→ φ × ψ defines
a Q-linear isomorphism
SCm ⊗ SCn ∼
= scf Um × Un
which allows us to identify SCm ⊗ SCn with the vector space scf(Um × Un ) consisting
of all superclass functions defined on Um × Un .
If J ⊆ [n] and |J| = m, then UJ/J c is an algebra subgroup of Un , and so we can
define a Q-linear map
J
ResUUnm ×Un−m : SCn → SCm ⊗ SCn−m
SUPERCHARACTERS AND SET PARTITION COMBINATORICS
53
by the rule
J
ResUUnm ×Un−m (φ) = ResUUnJ/J c (φ) ◦ st−1
J/J c
for all φ ∈ SCn ; we refer to this operation as the J-restriction from Un to Um × Un−m .
Of course, being a Q-linear map, theJ-restriction
is uniquely determined by the
images J ResUUnm ×Un−m (χπ ) on the Q-basis χπ : π ⊢ [n] consisting of all supercharacters
of Un . However, for a combinatorial description
of the Hopf
algebra structure of SC it
is more convenient to work with the Q-basis κπ : π ⊢ [n] consisting of all superclass
characteristic functions; in this situation we have the following result.
Lemma 4.5. If J ⊆ [n], |J| = m, and π ⊢ [n], then
eπ ∈ uJ/J c , in which case we have
J
ResUUnm ×Un−m (κπ ) = 0 unless
ResUUnm ×Un−m (κπ ) = κπI ⊗ κπI c
and I = i ∈ [ℓ(π)] : Bi ⊆ J .
J
where π = B1 /B2 / . . . /Bℓ(π)
Proof. By the definition, J ResUUnm ×Un−m (κπ ) is a superclass function of Um × Un−m ,
and hence it is a Q-linear combination of the superclass characteristic functions of
Um × Un−m . If σ ⊢ [m] and τ ⊢ [n − m] are arbitrary, then the superclass function
κσ ⊗ κτ ∈ scf(Um × Un−m ) = SCm ⊗ SCn−m appears in this linear combination with
a nonzero coefficient if and only if κπ (st−1
J/J c (gσ/τ )) 6= 0 where gσ/τ = 1 + eσ/τ is as in
(3l). Since
−1
−1
st−1
,
J/J c (gσ/τ ) = 1 + stJ/J c (eσ/τ ) = 1 + est
c (σ/τ )
J/J
we see that
κπ (st−1
J/J c (gσ/τ ))
6= 0 if and only if eπ ∈ uJ/J c and
−1
−1
π = st−1
J/J c (σ/τ ) = stJ (σ) ∨ stJ c (τ ).
Finally, if eπ ∈ uJ/J c , then
κπ (st−1
J/J c (gσ/τ )) = δπ,st−1 c (σ/τ ) = δstJ/J c (π),σ/τ ,
J/J
and this implies that
J
ResUUnm ×Un−m (κπ ) = κstJ/J c (π) = κπI ⊗ κπI c .
The result follows.
On the other hand, given any J ⊆ [n], we recall that the operation of superinduction
SIndUUnJ/J c : SC(UJ/J c ) → SC(Un )
associates with every φ ∈ scf(UJ/J c ) the superclass function SIndUUnJ/J c (φ) ∈ SCn defined
by
X
1
SIndUUnJ/J c (φ)(g) =
φ(1 + h(g − 1)k)
|UJ/J c |2
h,k∈Un
1+h(g−1)k∈UJ/J c
for all g ∈ Un . We also recall from Theorem 2.24 that
SIndUUnJ/J c (φ), ψ Un = φ, ResUUnJ/J c (ψ) U
J/J c
for all φ ∈ SC(UJ/J c ) and all ψ ∈ SC(Un ).
Now, if m, n ∈ N0 and J ⊆ [m + n] with |J| = m, then we may define the Q-linear
map
Um+n
J
SIndUm
×Un : SCm ⊗ SCn → SCm+n
54
CARLOS A. M. ANDRÉ
by the rule
J
U
U
m+n
m+n
SIndUm
×Un (φ) = SIndUJ/J c (φ ◦ stJ/J c )
for all φ ∈ SCm ⊗ SCn . We refer to this operation as the J-superinduction from Um × Un
to Um+n .
The following result is an easy consequence of Theorem 2.24.
Proposition 4.6. If m, n ∈ N0 and J ⊆ [m + n] is such that |J| = m, then
J
J
Um+n
Um+n
SIndUm
×Un (φ), ψ Um+n = φ, ResUm ×Un (ψ) Um ×Un
for all φ ∈ SCm ⊗ SCn and all ψ ∈ SCm+n .
Proof. Using Theorem 2.24, we deduce that
J
Um+n
Um+n
SIndUm
×Un (φ), ψ Um+n = SIndUJ/J c (φ ◦ stJ/J c ), ψ Um+n
Um+n
= φ ◦ stJ/J c , ResUJ/J
(ψ)
c
UJ/J c
Um+n
= φ, ResUJ/J c (ψ) ◦ st−1
J/J c Um ×Un
J
Um+n
= φ, ResUm
×Un (ψ) Um ×Un ,
as required.
On the basis of superclass characteristic functions, we obtain:
Lemma 4.7. If m, n ∈ N0 and J ⊆ [m + n] is such that |J| = m, then
J
|Kσ ||Kτ ||Um+n |
−1
κ −1
|Kst−1 (σ)∨st−1c (τ ) ||Um ||Un | stJ (σ)∨stJ c (τ )
U
m+n
SIndUm
×Un (κσ ⊗ κτ ) =
J
J
for all σ ⊢ [m] and all τ ⊢ [n].
Um+n
Proof. Let π ⊢ [m + n] be such that J SIndUm
×Un (κσ ⊗ κτ ), κπ Um+n 6= 0. By the
previous proposition and by Lemma 4.5, we deduce that
Um+n
κσ ⊗ κτ , J ResUm
×Un (κπ ) Um ×Un = κσ ⊗ κτ , κπI ⊗ κπI c Um ×Un 6= 0
where π = B1 /B2 / . . . /Bℓ(π) and I = i ∈ [ℓ(π)] : Bi ⊆ J . Therefore, we must have
κσ = κπI and κτ = κπI c , and so
−1
−1
π = st−1
J/J c (σ/τ ) = stJ (σ) ∨ stJ c (τ ).
It follows that
J
U
m+n
SIndUm
×Un (κσ ⊗ κτ ) = c κst−1 (σ)∨st−1
c (τ )
J
J
for some c ∈ Q. Now, it is easy to check that
κst−1 (σ)∨st−1c (τ ) , κst−1 (σ)∨st−1c (τ ) Um+n ,
J
J
J
J
and thus
J
U
m+n
SIndUm
×Un (κσ ⊗ κτ ), κst−1 (σ)∨st−1
c (τ )
J
J
Um+n
=c
|Kst−1 (σ)∨st−1c (τ ) |
J
J
|Um+n |
On the other hand, the argument above implies that
J
Um+n
=
κ
⊗
κ
,
κ
⊗
κ
SIndUm
σ
τ
σ
τ
(τ
)
×Un (κσ ⊗ κτ ), κst−1 (σ)∨st−1
c
Um+n
Um ×Un
J
.
J
|Kσ | |Kτ |
= κσ , κσ Um κτ , κτ Un =
,
|Um | |Un |
SUPERCHARACTERS AND SET PARTITION COMBINATORICS
and so c =
|Kσ ||Kτ ||Um+n |
. The proof is complete.
|Kσ/τ ||Um ||Un |
55
It follows that if we define
κ∗π =
(4k)
|Un |
κπ ∈ SCn
|Kπ |
for all π ⊢ [n] and all n ∈ N, then
(4l)
J
U
m+n
∗
∗
∗
SIndUm
×Un (κσ ⊗ κτ ) = κst−1 (σ)∨st−1
c (τ )
J
J
for
∗all σ ⊢ [m],
all τ ⊢ [n] and all m, n ∈ N0 . We observe that for every n ∈ N the set
κπ : π ⊢ [n] is a Q-basis of SCn , and that
∗
κπ , κπ′ = δπ,π′
for all π, π ′ ⊢ [n]; hence, the Q-basis κ∗π : π ⊢ [n] of SCn is dual to κπ : π ⊢ [n] with
respect to the Frobenius scalar product.
The next two operations are relevant for the description of the Hopf algebra structure
of SC, and arise naturally as follows. Given any m, n ∈ N0 , the subgroup Um × Un of
Um+n has a natural normal complement, namely the normal (abelian) subgroup
1 a
U(m,n) =
: a ∈ Mm×n (k)
0 1
where Mm×n (k) denotes the k-vector space consisting of all m × n matrices with
coefficients in k. Thus, there is a surjective homomorphism η : Um+n → Um × Un
satisfying η 2 = η; in fact, η fixes the subgroup Um × Un and has kernel ker(η) = U(m,n) .
On the one hand, for all m, n ∈ N0 , we define the inflation from Um × Un to Um+n
to be the map
Um+n
Inf Um
×Un : SCm ⊗ SCn → SCm+n
given by
U
m+n
Inf Um
×Un (φ) = φ ◦ η
for all φ ∈ SCm ⊗ SCn . On supercharacters, the inflation map is particularly nice, and
is given combinatorially by
U
m+n
σ
τ
σ/τ
Inf Um
×Un (χ × χ ) = χ
for all σ ⊢ [m] and all τ ⊢ [n]. On superclass characteristic functions, we have the
following.
Lemma 4.8. If m, n ∈ N0 , σ ⊢ [m] and τ ⊢ [n], then
X
Um+n
Inf Um
×Un (κσ ⊗ κτ ) =
κπ .
π⊢[m+n]
π∧(1̂m /1̂n )=σ/τ
Proof. For any π ⊢ [m + n], we have
U
m+n
Inf Um
×Un (κσ ⊗ κτ )(gπ ) = (κσ ⊗ κτ )(τ (gπ )).
The result follows because (κσ ⊗ κτ )(η(gπ )) = 0 unless π ∧ (1̂m /1̂n ) = σ/τ , in which
case (κσ ⊗ κτ )(η(gπ )) = 1.
56
CARLOS A. M. ANDRÉ
By the way of example, we have
b
b
b
1 2 3
·
b
b
b
b
1 2 3 4
=
b
b
b
b
b
b
+
b
1 2 3 4 5 6 7
+
b
b
b
b
b
b
b
b
b
b
b
b
+
b
1 2 3 4 5 6 7
b
1 2 3 4 5 6 7
+
b
b
b
b
b
b
b
b
b
b
b
+
b
1 2 3 4 5 6 7
b
b
1 2 3 4 5 6 7
+
b
b
b
b
b
b
b
b
b
b
b
b
b
1 2 3 4 5 6 7
b
1 2 3 4 5 6 7
where for simplicity we write π for κπ .
On the other hand, for every n ∈ N and every 0 ≤ k ≤ n, we define the deflation
from Un to Uk × Un−k to be the map
Def UUnk ×Un−k : SCn → SCk ⊗ SCn−k
given by
Def UUnk ×Un−k (ψ)(g) =
1
| ker(η)|
for all ψ ∈ SCn and all g ∈ Uk × Un−k .
X
ψ(h)
h∈η−1 (g)
Proposition 4.9. If m, n ∈ N0 , φ ∈ SCm ⊗ SCn and ψ ∈ SCm+n , then
Um+n
Um+n
Inf Um ×Un (φ), ψ Um+n = φ, Def Um
×Un (ψ) Um ×Un .
Proof. We evaluate
Um+n
Inf Um ×Un (φ), ψ Um+n = φ ◦ η, ψ Um ×Un =
=
1
X
|Um+n | h∈U
m ×Un
1
X
φ(η(g))ψ(g)
|Um+n | g∈U
m+n
X
φ(η(hk))ψ(hk)
k∈U(m,n)
X
X
1
φ(η(h))ψ(hk)
|Um × Un | |U(m,n) | h∈U ×U k∈U
m
n
(m,n)


X
X
1
1
=
ψ(hk)
φ(h) 
|Um × Un | h∈U ×U
|U(m,n) | k∈U
=
m
n
(m,n)
X
1
Um+n
φ(h) Def Um
×Un (ψ)(h)
|Um × Un |
h∈Um ×Un
Um+n
= φ, Def Um ×Un (ψ) Um ×Un ,
=
as required.
Lemma 4.10. If n ∈ N0 and π ⊢ [m + n], then
U
m+n
∗
∗
∗
Def Um
×Un (κπ ) = κst[m] (π∧(1̂m /0̂n )) ⊗ κst[m]c (π∧(0̂m /1̂n )) .
Um+n
∗
Proof. Let σ ⊢ [m] and τ ⊢ [n] be such that Def Um
×Un (κπ ), κσ ⊗ κτ Um ×Un 6= 0. Then,
by the previous proposition we have
∗
Um+n
Um+n
∗
κπ , Inf Um
×Un (κσ ⊗ κτ ) Um+n = Def Um ×Un (κπ ), κσ ⊗ κτ Um ×Un 6= 0,
and so by Lemma 4.8 we conclude that σ/τ = π ∧ (1̂m /1̂n ). Since 1̂m /1̂n ⊢ [m + n] has
blocks [m] and [m]c = [m+n]\[m], the blocks of π∧(1̂m /1̂n ) are the intersections B∩[m]
and B ∩ [m]c where B is a block of π; more precisely, if π has blocks B1 , . . . , Bℓ(π) ,
SUPERCHARACTERS AND SET PARTITION COMBINATORICS
57
then the blocks of π ∧ (1̂m /1̂n ) are B1 ∩ [m], . . . , Bℓ(π) ∩ [m] and B1 ∩ [m]c , . . . , Bℓ(π) ∩
[m]c (where we remove the empty intersections). From this we see that σ has blocks
B1 ∩ [m], . . . , Bℓ(π) ∩ [m], and that τ has blocks st[m]c (B1 ∩ [m]c ), . . . , st[m]c (Bℓ(π) ∩ [m]c ).
Equivalently, we have
σ = st[m] (π ∧ (1̂m /0̂n )) and τ = st[m]c (π ∧ (0̂m /1̂n )),
and this implies that
U
m+n
∗
∗
∗
Def Um
×Un (κπ ) = c κst[m] (π∧(1̂m /0̂n )) ⊗ κst[m]c (π∧(0̂m /1̂n ))
for some c ∈ Q. It follows that
∗
Um+n
∗
∗
Def Um
(κ
),
κ
⊗
κ
=
c
κ
⊗
κ
,
κ
⊗
κ
=c
σ
τ
σ
τ
π
σ
τ
×Un
Um ×Un
Um ×Un
where we write σ = st[m] (π ∧ (1̂m /0̂n )) and τ = st[m]c (π ∧ (0̂m /1̂n )). On the other hand,
the argument above implies that
∗
Um+n
∗
c = Def Um
×Un (κπ ), κσ ⊗ κτ Um ×Un = κπ , κπ Um+n = 1,
and this completes the proof.
4.4. The Hopf algebra structure of SC. This subsection defines explicitly the Hopf
structure on SC from a representation theoretic point of view. We then work out the
combinatorial consequences of these rules, and will eventually prove that SC is isomorphic to the Hopf algebra NCSym(X) consisting of symmetric functions on a countable
infinite set X = {x1 , x2 , x3 , . . .} of non-commuting variables as introduced by Rosas
and Sagan in [36].
There are two distinct natural ways of defining an associative product on SC. On the
one hand, given φ ∈ SCm and ψ ∈ SCn , it is tempting to define the product φ · ψ ∈ SC
using superinduction:
U
m+n
φ · ψ = SIndUm
×Un (φ ⊗ ψ).
(4m)
U
m+n
Since SIndUm
×Un =
[m]
U
m+n
SIndUm
×Un , (4l) implies that
U
∗
m+n
∗
∗
∗
SIndUm
×Un (κσ ⊗ κτ ) = κst−1 (σ)∨st−1 c (τ ) = κσ/τ
[m]
[m]
for all σ ⊢ [m] and all τ ⊢ [n]. Comparing this with (4i) we obtain the following.
Proposition 4.11. There is an isomorphism of graded Q-algebras ϕ : SC → NCSym
defined by ϕ(κ∗π ) = pπ for all π ⊢ [n] and all n ∈ N.
Remark 4.12. This result raises the following questions.
(1) Does the Hopf algebra structure of NCSym transfer in a representation theoretic
way
to SC?
(2) Is pπ : π ⊢ [n] the correct choice of the Q-basis of NCSymn ?
In fact, using the isomorphism ϕ : SC → NCSym and recalling (4j), it is possible to
define a coproduct ∆ : SC → SC ⊗ SC in an obvious way by the rule
X
κ∗πI ⊗ κ∗πI c
(4n)
∆(κ∗π ) =
I⊆[ℓ(π)]
for all π ⊢ [n] and all n ∈ N. Notice that ∆ is defined in order to guarantee that
∆ ◦ ϕ = (ϕ ⊗ ϕ) ◦ ∆,
58
CARLOS A. M. ANDRÉ
that is, that the diagram
∆
SC
SC ⊗ SC
/
ϕ
ϕ⊗ϕ
NCSym ⊗ NCSym
/
NCSym
∆
is commutative. However, it is not clear how to translate (4n) into the representation
theoretic language. On the other hand, there are other Q-bases of NCSym satisfying (4i)
which can be used to define algebra isomorphisms SC ∼
= NCSym and may possibly have
a better and more appropriate interpretation in terms of superclass functions.
The structure of Hopf algebra on SC is not unique. For instance, there is a second
natural definition of a product on SC which uses inflation: given φ ∈ SCm and ψ ∈ SCn
for any m, n ∈ N, we define
U
m+n
φ · ψ = Inf Um
×Un (φ ⊗ ψ).
(4o)
We follow the above procedure to obtain a different Hopf algebra structure on SC and
also a different Hopf isomorphism SC ∼
= NCSym. Recalling Lemma 4.8 and (4d) we
deduce the following.
Proposition 4.13. There is an isomorphism of graded Q-algebras ch : SC → NCSym
defined by ch(κπ ) = mπ for all π ⊢ [n] and all n ∈ N.
Now, using the isomorphism ch : SC → NCSym and recalling (4h), it is possible to
define a coproduct ∆ : SC → SC ⊗ SC by the rule
X
(4p)
∆(κπ ) =
κπI ⊗ κπI c
I⊆[ℓ(π)]
for all π ⊢ [n] and all n ∈ N. To illustrate, if π = 14/2/3 ⊢ [4], then
I {1, 2, 3} {1, 2} {1, 3} {2, 3} {1} {2} {3}
∅
πI 14/2/3 13/2 13/2
1/2
12
1
1
∅
,
πI c
∅
1
1
12
1/2 13/2 13/2 14/2/3
and thus
∆
b
b
b
b
1 2 3 4
=
b
b
b
1 2 3
+
b
⊗∅ +2
4
⊗
+2
b
b
1 2
b
b
1 2
b
b
b
1 2 3
b
1
⊗
⊗
b
b
1
b
b
1 2 3
⊗
+
1 2
+ ∅⊗
b
where for simplicity we write σ for κσ .
Notice that as before ∆ is defined in order to guarantee that
b
b
b
b
1 2
b
b
b
1 2 3 4
∆ ◦ ch = (ch ⊗ ch) ◦ ∆,
that is, that the diagram
SC
∆
/
SC ⊗ SC
ch ⊗ ch
ch
NCSym
/
∆
NCSym ⊗ NCSym
is commutative. In this situation, it possible to translate (4p) into the representation
theoretic language; indeed, as an immediate consequence of Lemma 4.5 we obtain the
following.
SUPERCHARACTERS AND SET PARTITION COMBINATORICS
59
Proposition 4.14. If π ⊢ [n] for any n ∈ N, then
X
J
ResUUn|J | ×U|J c | (κπ ).
∆(κπ ) =
J⊆[n]
Therefore, by linear extension we conclude that the coproduct ∆ : SC → SC ⊗ SC is
given by the rule
X
J
ResUUn|J | ×U|J c | (φ)
(4q)
∆(φ) =
J⊆[n]
for all φ ∈ SCn and all n ∈ N, and this concludes the proof of the following result.
Theorem 4.15. With respect to the product (4o) and to the coproduct (4q), the Qvector space SC becomes a Hopf algebra where the identity is κ∅ ∈ SC0 and the coidentity
ε : SC → C is obtained by taking the coefficient of κ∅ . Furthermore, the mapping
κ∗π 7→ mπ defines an isomorphism of Hopf algebras ψ : SC → NCSym.
In the following, we discuss the dual Hopf algebra SC∗ . As above, our central aim
is to provide representation theoretic interpretations of the product and coproduct of
SC∗ . We first discuss some general notions about dual Hopf algebras.
Let
M
An
A=
n∈N
be a graded Hopf algebra over an arbitrary field k. For every n ∈ N, let A∗n be the
dual k-vector space of An , and let A∗ denote the graded dual vector space
M
A∗ =
A∗n .
n∈N
(A∗ should not be confused with the dual vector space Homk (A, k) of A.)
For every ξ, ζ ∈ A∗ , we define the product ξ · ζ ∈ A∗ to be the composite map
ξ · ζ = (ξ ⊗ ζ) ◦ ∆
(4r)
where ∆ : A → A ⊗ A is the coproduct of A; thus, if a ∈ A, then
X
X
a′i ⊗ a′′i .
(ξ · ζ)(a) =
ξ(a′i )ζ(a′′i ) where ∆(a) =
i
i
It is routine to check that the mapping (ξ, ζ) 7→ ξ · ζ defines an associative product on
A∗ with respect to which A∗ becomes an associative k-algebra where the identity is
the coidentity ε : A → k of A.
On the other hand, for every ξ ∈ A∗ , we define the coproduct ∆∗ (ξ) ∈ A∗ ⊗ A∗ as
follows: we realise A∗ ⊗ A∗ as a subspace of the dual k-vector space (A ⊗ A)∗ in the
natural way, and define
(4s)
∆∗ (ξ)(a ⊗ b) = ξ(ab)
for all a, b ∈ A. Then, the mapping ξ 7→ ∆∗ (ξ) defines in fact a coproduct on A∗
which makes A∗ a graded Hopf algebra over k where the coidentity is the k-linear map
ε∗ : A∗ → k defined by ε∗ (ξ) = ξ(1) for all ξ ∈ A∗ .
We now particularise this construction to the graded Hopf Q-algebra SC. For every
n ∈ N0 , let SC∗n denote the dual Q-vector space SC∗n of SCn , and consider the graded
Q-vector space
M
SC∗ =
SC∗n .
n∈N
60
CARLOS A. M. ANDRÉ
We define the Frobenius scalar product ·, · on SC by extending naturally the usual
Frobenius
scalar product defined on SCn for every n ∈ N0 ; in particular, we ∗see that
Sm , Sn = 0 unless
every Q-linear map ξ ∈ SC as the
m = n. Then, we realise
∗
mapping φ 7→ ξ, φ , and hence we identify SC with SC (as Q-vector spaces).
Theorem 4.16. The product and coproduct of the Hopf algebra SC∗ are given as follows:
(1) If ξ ∈ SC∗m and ζ ∈ SC∗n for m, n ∈ N0 , then
X
Um+n
J
SIndUm
ξ·ζ =
×Un (ξ ⊗ ζ).
J⊆[m+n], |J|=m
(2) If ξ ∈ SC∗n for n ∈ N0 , then
∆∗ (ξ) =
X
Def UUnk ×Un−k (ξ).
0≤k≤n
Proof. Let r ∈ N and φ ∈ SCr be arbitrary. By (4r) we have ξ · ζ = (ξ ⊗ ζ) ◦ ∆, and
thus
X
(ξ · ζ)(φ) = (ξ ⊗ ζ)(∆(φ)) = ξ ⊗ ζ, ∆(φ) =
ξ ⊗ ζ, J ResUUr|J | ×U|J c | (φ) .
J⊆[r]
Since J ResUUr|J | ×U|J c | (φ) ∈ S|J| ⊗ S|J c | , we conclude that ξ ⊗ ζ, J ResUUr|J | ×U|J c | (φ) = 0
unless r = m + n and |J| = m, and so by Frobenius reciprocity we obtain
X
Um+n
(ξ · ζ)(φ) =
ξ ⊗ ζ, J ResUm
×Un (φ)
J⊆[m+n], |J|=m
=
X
J⊆[m+n], |J|=m
J
Um+n
SIndUm
(ξ
⊗
ζ),
φ
.
×Un
(1) follows (because r ∈ N and φ ∈ SCr are arbitrary).
For (2), let r, s ∈ N, φ ∈ SCr and ψ ∈ SCs be arbitrary. Then, by (4s) we have
U
(∆∗ (ξ))(φ ⊗ ψ) = ξ(φ · ψ) = ξ Inf Ur+s
(φ
⊗
ψ)
,
×U
r
s
and thus (∆∗ (ξ))(φ ⊗ ψ) = 0 unless n = r + s in which case we deduce that
(∆∗ (ξ))(φ ⊗ ψ) = ξ Inf UUnr ×Un−r (φ ⊗ ψ) = ξ, Inf UUnr ×Un−r (φ ⊗ ψ)
= Def UUnr ×Un−r (ξ), φ ⊗ ψ = Def UUnr ×Un−r (ξ)(φ ⊗ ψ).
Since r, s, φ and ψ are arbitrary, we conclude that
X
∆∗ (ξ) =
Def UUnr ×Un−r (ξ)
0≤r≤n
as required.
∗
n ∈ N, let κπ : π ⊢ [n] be
the Q-basis
of SC∗n which is dual to the Q-basis
For every ∗
∗
κπ : π ⊢ [n] of SCn ; hence, κπ (κπ′ ) = κπ , κπ′ = δπ,π′ for all π, π ′ ⊢ [n], and so
κ∗π =
|Un |
κπ
|Kπ |
for all π ⊢ [n] (see (4k)). The following result is an immediate consequence of (4l) and
Lemma 4.10.
SUPERCHARACTERS AND SET PARTITION COMBINATORICS
61
(1) If σ ⊢ [m] and τ ⊢ [n] for m, n ∈ N0 , then
X
κ∗σ · κ∗τ =
κ∗st−1 (σ)∨st−1c (τ ) .
Proposition 4.17.
J
J⊆[m+n], |J|=m
J
(2) If π ⊢ [n] for n ∈ N0 , then
X
∆∗ (κ∗π ) =
κ∗st[k] (π∧(1̂k /0̂n−k )) ⊗ κ∗st[k]c (π∧(0̂k /1̂n−k )) .
0≤k≤n
To illustrate, we have (we write π for κ∗π , so that σ · τ means κ∗σ · κ∗τ )
b
b
1 2
·
b
b
b
1 2 3
=
b
b
b
b
+
b
1 2 3 4 5
+
b
b
b
b
b
b
b
b
+
b
1 2 3 4 5
+
b
1 2 3 4 5
b
b
b
b
b
b
b
+
b
1 2 3 4 5
b
b
1 2 3 4 5
+
b
b
b
b
b
b
b
+
b
1 2 3 4 5
b
b
1 2 3 4 5
+
b
b
b
b
b
b
b
b
1 2 3 4 5
b
+
b
1 2 3 4 5
b
b
b
b
b
1 2 3 4 5
and
∆∗
b
b
b
b
1 2 3 4
=
b
b
b
b
1 2 3 4
+
b
1
⊗∅ +
⊗
b
b
b
1 2 3
b
b
b
1 2 3
⊗
+ ∅⊗
b
b
1
b
b
+
b
1 2 3 4
b
b
1 2
⊗
b
b
1 2
4.5. The subalgebra LSC. The Hopf algebra SC has a number of natural Hopf subalgebras. One of particular interest is the subspace spanned by the supercharacters χπ with
a linear constituent. In fact, the linear characters of Un are precisely the supercharacters χπ,φ where (π, φ) is a k× -coloured set partition such that D(π)
⊆ D([n]) where we
write [n] for the set partition 1̂n = 12 · · · n; notice that D([n]) = (i, i + 1) : 1 ≤ i < n .
In fact, we have the following result.
Theorem 4.18. If n ∈ N and χ is a character of Un , then χ is linear (that is, χ(1) = 1)
if and only if χ = χπ,φ for some (π, φ) ∈ Sn (k) such that D(π) ⊆ D([n]).
Proof. On the one hand, we have χπ,φ (1) = 1 for all (π, φ) ∈ Sn (k) with D(π) ⊆ D([n])
(see Theorem 3.8), and hence these supercharacters are at least q n−1 linear characters
of Un ; as usual, we set q = |k|. On the other hand, let Un′ = [Un , Un ] denote the
commutator subgroup of Un (which is generated by [g, h] = g −1h−1 gh for all g, h ∈ un ).
It is clear that every linear character χ of Un satisfies χ(Un′ ) = 1, and thus χ defines
a character of the quotient group Un /Un′ ; recall that Un′ is a normal subgroup of Un ,
and in fact it is the smallest normal subgroup with abelian quotient. It follows that
the number of linear characters of Un is not greater that |Un /Un′ |; indeed, it is a basic
fact that this number equals |Un/Un′ |. It is an easy exercise to show that Un′ = 1 + u′n
where u′n is the k-linear span of ei,j : (i, j) ∈ [[n]], j − i ≥ 2 , and thus the number of
linear characters of Un is not greater that |Un /Un′ | = q n−1 . The result follows.
We define LSC to be the Q-vector subspace of SC spanned by all supercharacters
X
1
χπ =
χπ,φ
(q − 1)d(π)
φ∈Colk (π)
62
CARLOS A. M. ANDRÉ
where q = |k|, n ∈ N0 , and π ⊢ [n] is such that D(π) ⊆ D([n]). Moreover,
for every
n ∈ N, we define LSCn to be the Q-vector subspace of SCn spanned by χπ : π ⊢ [n] ;
then, LSC is naturally graded as
M
LSC =
LSCn .
n∈N
Lemma 4.19. LSC is a (graded) Hopf subalgebra of SC.
Proof. If σ ⊢ [m] and τ ⊢ [n] for m, n ∈ N, then
U
m+n
σ
τ
σ/τ
χσ · χτ = Inf Um
.
×Un (χ · χ ) = χ
Since D(σ/τ ) ⊆ D([m + n]) whenever D(σ) ⊆ D([m]) and D(τ ) ⊆ D([n]), we conclude
that LSC is a (graded) subalgebra of SC. On the other hand, if n ∈ N and π ⊢ [n] is
such that D(π) ⊆ D([n]), then ResUUnJ/J c (χπ ) is a linear character of UJ/J c , and thus
ResUUnk ×Un−k (χπ ) is a linear character of Uk × Un−k for all J ⊆ [n] with |J| = k. Since
every linear character of Uk × Un−k is a product χ × χ′ where χ is a linear character of
Uk and χ′ is a linear character of Un−k , it follows that J ResUUnk ×Un−k (χπ ) ∈ LSCk ⊗LSCn−k
for all J ⊆ [n] with |J| = k, and thus
M
X
J
LSCk ⊗ LSCn−k
ResUUn|J | ×U|J c | (χπ ) ∈
∆(χπ ) =
J
0≤k≤n
J⊆[n]
which proves that LSC is a graded subcoalgebra of SC. The proof is complete.
In [24], the authors introduced the Q-algebra Sym of formal non-commutative symmetric functions which is just the free associative Q-algebra QhhΛ1 , Λ2 , Λ3 , . . .ii generated by an infinite sequence of indeterminates Λ1 , Λ2 , Λ3 , . . . over Q (which the authors
call the elementary symmetric functions). This algebra is graded by the weight function ω(Λn ) = n for all n ∈ N 7, and the homogeneous component of weight n is denoted
by Symn ; hence,
M
Sym =
Symn .
n∈N
It is a Hopf algebra with respect to the coproduct ∆ : Sym → Sym ⊗ Sym defined by
X
∆(Λn ) =
Λk ⊗ Λn−k
0≤k≤n
for all n ∈ N (see [24, Proposition 3.8]). In fact, Sym is freely generated by the
power sums symmetric functions of the first kind Ψ1 , Ψ2 , Ψ3 , . . . ([24, Definition 3.1
and Proposition 3.3]), with respect to which the coproduct is given by
(4t)
∆(Ψn ) = Ψn ⊗ 1 + 1 ⊗ Ψn
for all n ∈ N.
Proposition 4.20. The Hopf subalgebra LSC of SC is isomorphic to the Hopf algebra
Sym.
7In
the usual setting the grading is given by the weight function ω(Λn ) = 1; the reason here is that
the indeterminate Λn should be regarded as an elementary symmetric function.
SUPERCHARACTERS AND SET PARTITION COMBINATORICS
63
Proof. By Theorem 3.8, if π ⊢ [n] for n ∈ N, then χπ , κ[n] = 0 unless D(π) ⊆ D([n]),
and thus κ[n] ∈ LSC. Furthermore, with respect to the partial order ≤ on set partitions,
the set of products
κ[n1 ] κ[n2 ] · · · κ[nr ] : n1 + n2 + · · · + nr = n, r ∈ N
have an upper-triangular decomposition in terms of the characteristic functions κπ for
π ⊢ [n] and n ∈ N0 . It follows that the elements κ[n] are algebraically independent in
LSC, and hence LSC contains the free Q-algebra Qhhκ[1] , κ[2] , κ[3] , . . .ii.
Now, note that every π ⊢ [n] with D(π) ⊆ D([n]) has blocks of the form [n1 ], [n2 ]+n1 ,
. . . , [nr ] + (n1 + · · · + nr ) where 1 ≤ n1 , . . . , nr ≤ n are such that n1 + · · · + nr = n,
and so for every n ∈ N the vector subspace LSCn has dimension
dimC LSCn = κ[n ] κ[n ] · · · κ[nr ] : n1 + n2 + · · · + nr = n, r ∈ N .
1
2
This implies that
LSC = Qhhκ[1] , κ[2] , κ[3] , . . .ii.
Since ℓ([n]) = 1, (4p) implies that
∆(κ[n] ) = κ∅ ⊗ κ[n] + κ[n] ⊗ κ∅ = 1 ⊗ κ[n] ⊗ κ[n] ⊗ 1,
and thus the desired isomorphism is defined by the mapping κ[n] 7→ Ψn (by (4t)).
(m)
Remark 4.21. For every m, n ∈ N, let SCn be the Q-vector subspace
of SCn spanned by
all the supercharacters χπ where π ⊢ [n] is such that D(π) ⊆ (i, j) ∈ [[n]] : j −i ≤ m .
Then,
M
SCn(m)
SC(m) =
n∈N
is a Hopf subalgebra of SC, and this gives an (unexplored) filtration of Hopf algebras
which interpolate between LSC = SC(1) and SC.
Exercises.
4.1. Let (i, j) ∈ [[n]], and consider the supercharacter χ{ij} of Un ; recall that {ij}
denotes the unique set partition π ⊢ [n] such that D(π) = {(i, j)}. For every J ⊆ [n],
determine:
(1) the decomposition of ResUUnJ (χ{ij} ) as a linear combination of supercharacters.
[Hint. Consider first the case J = [n] \ {k} for some 1 ≤ k ≤ n, and then the
case where J = {k, k + 1, . . . , l} for (k, l) ∈ [[n]].]
(2) the decomposition of SIndUUnJ (χ{ij} ) as a linear combination of supercharacters.
4.2. Let J ⊆ [n], and consider the group isomorphism stJ : UJ → Um where m = |J|;
note that every supercharacter of UJ has the form χσ ◦ st−1
J where σ ⊢ [m]. Prove that:
(1) If
X
X
Un
π
π
(χ
)
=
bσ,π (χσ ◦ st−1
)
=
a
χ
and
Res
SIndUUnJ (χσ ◦ st−1
σ,π
UJ
J ),
J
σ⊢[m]
π⊢[n]
then q crs(π) aσ,π = q crs(σ) bσ,π for all σ ⊢ [n] and all π ⊢ [m].
(2) If J = [m], then
X
(q − 1)d(π) q − crs(π) χπ
SIndUUn[m]/[m]c (1) =
π⊢[m]
(i,j)∈D(π) =⇒ i6∼j
where 1 denotes the unit character of U[m]/[m]c , and i ∼ j means that i and j
are in the same block of J.
64
CARLOS A. M. ANDRÉ
4.3. ([44, Corollary 4.9]) Let SGm,n be the set consisting of all 0 − 1 matrices of size
m × n having at most one 1 in every row and every column. For every w ∈ SGm,n ,
define
ones(w) = | (i, j) ∈ [m] × [n] : wi,j = 1 |,
sow(w) = | (i, j) ∈ [m] × [n] : wi,j = 0, ∃k<i wk,j = 1 or ∃l>j wil = 0 |
For example, if


0 1 0 0
w =  0 0 0 1 ,
0 0 0 0
then ones(w) = 2 and sow(w) = 6. Prove that the following hold for all m, n ∈ N;
here, t is an indeterminate.
X
(t − 1)ones(w) tsow(w) .
(1) tmn =
(2) 0 =
w∈SGm,n
X
(−1)w1,n (t − 1)ones(w) tsow(w) .
w∈SGm,n
[Hint. Use the previous exercise.]
References
[1] M. Aguiar, C.A.M. André, C. Benedetti, N. Bergeron, Z. Chen, P. Diaconis, A. Hendrickson,
S. Hsiao, I.M. Isaacs, A. Jedwab, K. Johnson, G. Karaali, A. Lauve, T. Le, S. Lewis, H. Li, K.
Magaard, E. Marberg, J-C. Novelli, A. Pang, F. Saliola, L. Tevlin, J-Y. Thibon, N. Thiem, V.
Venkateswaran, C.R. Vinroot, N. Yan, M. Zabrocki, Supercharacters, symmetric functions in
noncommuting variables, and related Hopf algebras, Adv. Math. 299, 2310-2337 (2012).
[2] M. Aguiar, N. Bergeron & F. Sottile, Combinatorial Hopf Algebras and generalized DehnSommerville relations, Compositio Math. 142, 1-30 (2006).
[3] M. Aguiar & S. Mahajan, Coxeter groups and Hopf algebras, Amer. Math. Soc. Fields Inst.
Monogr. 23, 2006.
[4] M. Aguiar & S. Mahajan, Monoidal functors, species and Hopf algebras, CRM Monograph
Series, 29, Amer. Math. Soc., Providence, 2010.
[5] C.A.M. André, Basic characters of the unitriangular group, J. Algebra 175, 287-319 (1995).
[6] C.A.M. André, Basic sums of coadjoint orbits of the unitriangular group, J. Algebra 176,
959-1000 (1995).
[7] C.A.M. André, Irreducible characters of finite algebra groups, in Matrices and Group Representations, Coimbra, 1998. Textos Mat. Sér. B 19 (1999), 65-80.
[8] C.A.M. André, The basic character table of the unitriangular group, J. Algebra 241, 437-471
(2001).
[9] C.A.M. André, Basic characters of the unitriangular group (for arbitrary primes), Proc. Amer.
Math Soc. 130, 1943-1954 (2002).
[10] C.A.M. André & A.P. Nicolás, Supercharacters of the adjoint group of a finite radical ring, J.
Group Theory 11 (5), 709-746 (2008).
[11] D.J. Anick, On the homogeneous invariants of a tensor algebra, in “Algebraic Topology: Proceedings of the International Conference held March 21-24, 1988”, Mark Mahowald and Stewart
Priddy eds., Contemporary Mathematics, Vol. 96, American Math. Society, Providence, 1989.
[12] N. Bergeron, C. Hohlweg, M. Rosas & M. Zabrocki, Grothendieck bialgebras, partition lattices,
and symmetric functions in noncommutative variables, Electron. J. Combin. 13, no. 1, Research
Paper 75, 19 pp (2006).
[13] N. Bergeron, C. Reutenauer, M. Rosas & M. Zabrocki, Invariants and coinvariants of the
symmetric groups in noncommuting variables, Canad. J. Math. 60, no. 2, 266-296 (2008).
[14] N. Bergeron & N. Thiem, A supercharacter table decomposition via power-sum symmetric functions, Int. J. of Algebra and Computation 23, no. 4, 763-778 (2013).
SUPERCHARACTERS AND SET PARTITION COMBINATORICS
65
[15] G.M. Bergman & P.M. Cohn, Symmetric elements in free powers of rings, J. London Math.
Soc. (2) 1, 525-534 (1969).
[16] J.L. Brumbaugh, M. Bulkow, P.S. Fleming, L.A. Garcia, S.R. Garcia, G. Karaali, M. Michal,
& A.P. Turner, Supercharacters, exponential sums, and the uncertainty principle. (Available on
the electronic archives as arXiv:1208.5271)
[17] V. Chari & A. Pressley, A guide to quantum groups, Cambridge University Press, Cambridge,
1994.
[18] P. Diaconis & I.M. Isaacs, Supercharacters and Superclasses for Algebra Groups, Trans. Amer.
Math. Soc. 360, no. 5, 2359–2392 (2007).
[19] V.G. Drinfeld, Quantum groups, in “Proceedings ICM”, Berkeley, Amer. Math. Soc. 798-820
(1987).
[20] P.S. Fleming, S.R. Garcia & G. Karaali, Classical Kloosterman sums: representation theory,
magic squares, and Ramanujan multigraphs, J. Number Theory 131, no. 4, 661-680 (2011).
[21] C.F. Fowler, S.R. Garcia & G. Karaali, Ramanujan sums as supercharacters. (Available on the
electronic archives as arXiv:1201.1060)
[22] W. Fulton, Young Tableaux, London Mathematical Society Student Texts 35, Cambridge University Press, Cambridge, 1999.
[23] D. Gebhard & B. Sagan, A chromatic symmetric function in noncommuting variables, J. Algebraic Combin. 13, 227-255 (2001).
[24] I.M. Gelfand, D. Krob, A. Lascoux, B. Leclerc, V. S. Retakh & J.-Y. Thibon, Noncommutative
symmetric functions, Adv. Math. 112, 218-348 (1995).
[25] F. Hivert, J.-C. Novelli & J.-Y. Thibon, Commutative combinatorial Hopf algebras, J. Algebraic
Combin. 28, no. 1, 65-95 (2008).
[26] I.M. Isaacs, Character Theory of Finite Groups, Dover Publications Inc., New York, 1976.
[27] I.M. Isaacs, Characters of groups associated with finite algebras, J. Algebra 177, 708-730 (1995).
[28] I.M. Isaacs & D. Karagueuzian, Conjugacy in groups of upper triangular matrices, J. Algebra
202, 704-711 (1998).
[29] I.M. Isaacs & D. Karagueuzian, Erratum: Conjugacy in groups of upper triangular matrices,
J. Algebra 208, 722 (1998).
[30] S.A. Joni & G.C. Rota, Coalgebras and bialgebras in combinatorics, Stud. Appl. Math. 61,
93-139 (1979).
[31] V.K. Kharchenko, Algebras of invariants of free algebras, Algebra i Logika 17, 478-487 (1978)
(Russian); Algebra and Logic 17, 316-321 (1978) (English translation).
[32] A.A. Kirillov, Variations on the triangular theme, in Lie groups and Lie algebras: E.B. Dynkin’s
Seminar, 43-73, Amer. Math. Soc. Transl. Ser. 2 169, Providence, 1995.
[33] J.H. van Lindt & R.M. Wilson, A Course in Combinatorics, Cambridge University Press,
Cambridge, 2001.
[34] I.G. MacDonald, Symmetric functions and Hall polynomials, 2nd edition, Oxford University
Press, Oxford, 1995.
[35] S. Montgomery, Hopf algebras and their actions on rings, CBMS Regional Conference Series
in Mathematics 82, Washington DC, 1993.
[36] M. Rosas & B. Sagan, Symmetric functions in noncommuting variables, Trans. Amer. Math.
Soc. 358, 215-232 (2006).
[37] B. Sagan, The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric
Functions, 2nd edition, Springer-Verlag, New York, 2001.
[38] J. Sangroniz, Characters of algebra groups and unitriangular groups. Finite Groups 2003, 335349. de Gruyter, Berlin, 2004.
[39] S. Shnider & S. Sternberg, Quantum groups: From coalgebras to Drinfeld algebras, a guided
tour, Graduate Texts in Mathematical Physics II, International Press, Cambridge, 1993.
[40] R.P. Stanley, A symmetric function generalization of the chromatic polynomial of a graph,
Advances in Math. 111, 166-194 (1995).
[41] R.P. Stanley, Enumerative Combinatorics, Volume 1, Cambridge University Press, Cambridge,
1997.
[42] R.P. Stanley, Graph Colorings and related symmetric functions: ideas and applications: A
description of results, interesting applications, and notable open problems, Selected papers in
honor of Adriano Garsia (Taormina, 1994) Discrete Math. 193, 267-286 (1998).
66
CARLOS A. M. ANDRÉ
[43] R.P. Stanley, Enumerative Combinatorics, Volume 2, Cambridge University Press, Cambridge,
1999.
[44] N. Thiem, Branching rules in the ring of superclass functions of unipotent upper-triangular
matrices, J. Algebraic Combin. 31, no. 2, 267-298 (2010).
[45] N. Yan, Representation theory of the finite unipotent linear groups, Unpublished Ph.D. Thesis,
Department of Mathematics, University of Pennsylvania, 2001.
[46] M.C. Wolf, Symmetric functions of noncommuting elements, Duke Math. J. 2, 626-637 (1936).
Departamento de Matemática, Faculdade de Ciências da Universidade de Lisboa,
Campo Grande, Edifício C6, Piso 2, 1749-016 Lisboa, Portugal and Centro de Estruturas Lineares e Combinatórias, Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal
E-mail address: [email protected]