Group Theory, Lattice Geometry, and Minkowski’s Theorem
Jason Payne
Physics and mathematics have always been inextricably interwoven- one’s development and
progress often hinges upon the other’s. It is that which motivates this primer on group theory
and lattice geometry in the hopes of presenting the mathematical tools refined by explorations into
a field known as algebraic number theory in a context more easily accessible to a physicist studying
lattice structure. The focus will be on clarity of exposition and a grounding in examples of the more
elusive concepts.
This paper is separated into two parts. First, we will introduce a selection of the vital foundational concepts in the theory of groups. Following this will be a basic introduction to lattice
geometry: bringing to light some crucial concepts like that of lattice vectors, the interplay between
(n-dimensional) lattices and the (n-dimensional) Torus Tn , fundamental domains, and convexity.
This will culminate in the proof of Minkowski’s Theorem which states that if a bounded, symmetric,
convex subset X of Rn satisfies the following condition on its volume,
V (X) > 2n V (D),
where D is the fundamental domain of an n-dimensional lattice L, then X must contain a non-zero
lattice vector. This theorem has far reaching implications in the algebraic theory of numbers, as well
as their geometry, and consequently in the lattice-entrenched theory of condensed matter physics.
It is the hope of the author that with this brief introduction the reader can involve themselves in a
deeper study of the theory of solids relying upon a rigorous mathematical formalism of the internal
geometry of crystals.
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I.
INTRODUCTION TO GROUP THEORY
We begin with a very brief review of the essential elements of group theory. Although group theory is, by itself,
a substantial and vital portion of modern mathematics, we will focus on the foundational concepts relevant to the
discussion that follows. In the interest of clarity, the more subtle details (for example, the potential existence of a
left-inverse that is not a right-inverse) will be neglected. Of course, all of this would be for naught without first
introducing the notion of a group.
Definition 1. Given a (possibly infinite) collection of objects G = {g1 , · · · , gn , · · · } coupled with an operation · :
G × G → G, we call G a group if it satisfies the following:
(i) (Closure) Given any two elements g, h ∈ G, we have that g · h ∈ G.
(ii) (Group Identity) There exists an element e ∈ G so that
g · e = e · g = g.
(∀g ∈ G)
e is called the identity element of G.
(iii) (Inverses) Given any g ∈ G there exists an element h ∈ G so that
g · h = h · g = e.
h is called the (two-sided) inverse of g and is denoted g −1 .
This is merely an abstraction of an idea with which we are all familiar. The focus of group theory is on examining
precisely how the general principles of arithmetic or multiplication apply in other situations – it is an analysis of the
blueprints of arithmetic and multiplicative structures. It should be pointed out that often times care is necessary in
describing a group, as the same set of elements can have multiple, distinct operations defined on it. A few examples
that should be familiar to the reader are given below.
Ex 1. Consider R with the usual notion of addition (or, more generally, Rn with the usual notion of vector addition).
The (additive) identity element is given by e = 0, and the (additive) inverse of a real number a is simply −a. Similarly,
one could use the real numbers with multiplication. This latter remark highlights what was said above – it is, in some
cases, important to signify the operation one is interested in. In the context of this example, one can achieve this by,
rather than simply referring to the group R, talking about the group (R, +) or the group (R, ·), where the first term in
the pair identifies the set of interest, and the second denotes the operation.
Ex 2. The physical Hilbert space associated with a quantum mechanical system (or, more generally, any vector space)
has contained within it a group structure – that of the states, given by kets |·i, and addition operation to generate
(potentially) new states. Indeed, recall that in the definition of the underlying vector space of a Hilbert space the
closure, identity, and inverse properties are included.
Ex 3. Consider S = {z ∈ C | |z| = 1} with · : S × S → S defined as the usual complex multiplication:
(a + ib) · (c + id) = (ac − bd) + i(ad + bc).
This is called the circle group and it is related to rotations.
Definition 2. Let (G, ·) and (H, +) be groups. Then a map f : G → H is called a (group) homomorphism if for
all g1 , g2 ∈ G we have
f (g1 · g2 ) = f (g1 ) + f (g2 ).
If, in addition, f is one-to-one, onto, or both then it is called a monomorphism, epimorphism, and isomorphism,
respectively.
One point of clarification is that the above operations · and + do not necessarily have any relation to the usual
notions of addition and multiplication – they simply signify that we (potentially) have two different operations in play
here. So what a homomorphism really is, is a map which, in a sense, preserves the operations of the group – it is a
map for which you can either operate on the arguments in G before applying the map, or apply the map to each of
the arguments separately and then operate in H on the results.
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Definition 3. Let H ⊂ G. Then H is called a subgroup of G if H satisfies (i) – (iii) above when considered as a
set with the same operation as G. That is to say that it is closed under the operation of G, it includes the identity
element of G, and h ∈ H implies that h−1 ∈ H, where h−1 is the same inverse as when h is considered as an element
of G.
An equivalent, and typically useful, characterization of the idea of a subgroup is that it is a subset H for which,
given any h1 , h2 ∈ H we also have that
h1 · h−1
2 ∈ H.
We now introduce two very special subgroups related to any given homomorphism.
Definition 4. Let f : G → H be a group homomorphism. Then the subsets
ker f = {g ∈ G | f (g) = eH }
and
im f = {h ∈ H | f (g) = h for some g ∈ G}
are called the kernel of f and the image of f , respectively.
Theorem 1. Let N < G; then the following conditions on N are equivalent:
(i) gN = N g,
∀g ∈ G;
(ii) gN g −1 ⊂ N,
∀g ∈ G;
(iii) gN g −1 = N,
∀g ∈ G.
A subgroup N which satisfies any of the above conditions is called a normal subgroup, and this is denoted by N ⊳ G.
Proof.
(i) ⇒ (ii): Suppose that gN = N g for all g ∈ G. This means that for a fixed g ∈ G and n ∈ N , there exists an element
n′ ∈ N so that
gn = n′ g.
Multiplying on the right by g −1 we have that
−1
= n′ ∈ N.
gng −1 = n′
gg
Repeating this for each n ∈ N and g ∈ G yields gN g −1 ⊂ N , as desired.
(ii) ⇒ (iii): Suppose that gN g −1 ⊂ N . To demonstrate equality we need to establish the reverse containment, i.e. show
N ⊂ gN g −1 . The key to this is to realize that (ii) applies to g −1 as well, which allows us to write, for any
n∈N
n = (gg −1 )n(gg −1 ) = g(g −1 ng)g −1 = gn′ g −1 ∈ gN g −1 ;
thus N ⊂ gN g −1 as well, so we have N = gN g −1 .
(iii) ⇒ (i): This is immediate – just multiply both sides of (iii) on the right by g:
(gN
g −1
)g = N g
⇒
gN = N g.
The above implications give us a closed loop, of sorts, of implications which establishes the equivalence of the three
claims:
(i)
(ii)
(iii)
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Having provided a few equivalent ways of thinking about normal subgroups, we are now able to introduce another
important concept in group theory – that of the quotient group.
Definition 5. Given a normal subgroup N ⊳ G, then we can define a new group, G/N , associated with N as the set
of all (left) cosets of N in G, that is
G/N = {gN | g ∈ G}.
The operation which turns this into a group is given by
(gN ) · (hN ) = (gh)N.
(g, h ∈ G)
An important technical detail to keep in mind is that in order for the set G/N to be a group, N must be a normal
subgroup. For an intuitive understanding of what the quotient group is, one can think of it as an identification – you
are, in some sense, taking all the elements of the subgroup N and identifying them with each other. You treat them
as if they are the same element in order to focus on the action N has with other elements of the group G.
Next, we point out that some of the most important theoretic tools group-theorists have at their disposal at that
of the so-called isomorphism theorems. We will only make use of the first one:
Theorem 2. Let ϕ : G → H be a homomorphism of groups. Then
(i) ker ϕ ⊳ G;
(ii) im(ϕ) < H;
(iii) (First Isomorphism Theorem) im(ϕ) ∼
= G/ ker(ϕ).
Proof.
(i) Let x ∈ ker ϕ and g ∈ G. By the definition of homomorphism we have
ϕ(gxg −1 ) = ϕ(g)ϕ(x)ϕ(g −1 ) = ϕ(g)eϕ(g)−1 = e;
thus gxg −1 ∈ ker ϕ. Since x was an arbitrary element of the kernel this implies that g(ker ϕ)g −1 ⊂ ker ϕ.
Therefore by Theorem 1(ii) we have that ker ϕ ⊳ G, as desired.
(ii) Let a, b ∈ im ϕ. Then there exist some g, h ∈ G so that ϕ(g) = a and ϕ(h) = b. However, this means that
ab−1 = ϕ(g)ϕ(h)−1 = ϕ(g)ϕ(h−1 ) = ϕ(gh−1 ) ∈ im ϕ;
hence im ϕ < H.
(iii) (Sketch) The situation we are dealing with can be codified by the following diagram:
G
ϕ
im ϕ
π
ϕ
e
G/ ker ϕ
where π : G → G/ ker ϕ is the quotient map and ϕ
e is the isomorphism we are looking for. The idea is to define
ϕ
e so that the above diagram is commutative, i.e. one can travel around the diagram along either of the two
paths (ϕ or ϕ
e ◦ π) with the same result. Therefore, we let ϕ
e : G/ ker ϕ → im ϕ be defined as
ϕ(g(ker
e
ϕ)) = ϕ(g).
Since this proof is included simply to illuminate a common approach in abstract algebraic problems through the
above definition, we will, for the sake of brevity, leave the remainder to the reader.
The interested reader is invited to explore the rich framework that group-theoretic investigations has illuminated
within modern mathematics. A particularly comprehensive and sophisticated treatise on the subject is given by [3];
alternatively, an intuitive, example-driven discussion can be found in [1].
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II.
INTRODUCTION TO LATTICE GEOMETRY
[Note: This section will closely follow [6]. Additional insight came from [4] and [5], as well as the author’s understanding of the subject based on a course in algebraic number theory using the text [2].]
The starting point for this section is the definition of a lattice – a concept closely related to, and occasionally
analogous to, that of a vector space. For simplicity, we will consider only lattice subsets of Rn .
Definition 6. Let {e1 , · · · , en } be a set of linearly independent vectors in Rn . Then the additive subgroup L < Rn
generated by e1 , · · · , en is called an n-dimensional lattice. This subgroup consists of all the elements v ∈ Rn which
can be written in the form
v=
n
X
nk ek .
(nk ∈ Z)
k=1
Associated with any lattice is a particular region in Rn , called the fundamental domain (we will denote this region
by D), which consists of all elements w ∈ Rn of the form
w=
n
X
ak ek .
(0 ≤ ak < 1)
k=1
This is what is known as the unit cell in the theory of solids.
We can go immediately from this definition to the following theorem:
Theorem 3. Let L be an n-dimensional lattice in Rn . The L is isomorphic to the n-dimensional torus Tn .
Sketch. We begin by viewing the generating vectors of the lattice e1 , · · · , en as a basis for Rn . This means, in
particular, that any vector v ∈ Rn can be represented as
v=
n
X
ak e k .
(ak ∈ R)
k=1
We can define a map φ : Rn → Tn by
φ(v) = (e2πia1 , e2πia2 , · · · , e2πian ).
Then we have immediately that φ is a homomorphism, and is onto. There are two important details to notice – the
first is that if v ∈ L then ak ∈ Z for all k, so φ(v) = (0, 0, · · · , 0), i.e. we have that
ker φ = L.
Secondly, since φ is onto we have that im φ = Tn ; hence, by the First Isomorphism Theorem for groups, we have an
isomorphism
Rn /L = Rn / ker φ ∼
= im φ = Tn ,
as desired.
This theorem may, on the face of it, appear to be more of a mathematical curiosity than a concept deeply rooted in
the study of lattices; however, it turns out, this theorem provides the connection between lattices and the second topic
of this section – geometry. One of the foundational objects of interest in any geometrical field of mathematics is that
of measure – how one can assign appropriate notions of length, area, volume, and their n-dimensional counterparts to
the elements of the topological space of interest. Fortunately, in the case of lattice geometry we have just established a
connection between the abstract concept of an additive subgroup of Rn and the more concrete (and well understood)
object known as a torus. It is through this correspondence that we can develop a suitable perspective on the meaning
of (n-dimensional) volume in a lattice, as seen in the following definition.
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Definition 7. Recall that the volume of a (measurable) subset X ⊆ Rn is given by integrating against the Lebesgue
measure (which for simple, well-behaved subsets like the torus is equivalent to the more familiar Riemann integral of
the volume element dx1 · · · dxn ):
Z
v(X) =
dx1 · · · dxn .
X
To extend this to the more indefinite idea of a lattice, we utilize the above correspondence in order to pullback the
notion of volume through φ. Thus, the volume of a subset X ⊆ D, where D is the fundamental domain of the lattice
L, is defined to be
Z
−1
v(X) = v(φ (X)) =
dx1 · · · dxn .
φ−1 (X)
The following theorem (stated without proof) provides a useful characterization of the relationship between the
volume of a subset X ⊆ Rn and that of a subset of the lattice L.
Theorem 4. Let φ : Rn → Tn be defined as in Theorem 3. If X is a bounded subset of Rn , v(X) exists, and
v(φ(X)) 6= v(X), then φ|X : X → Tn is not one-to-one.
Finally, before moving onto the primary result presented in this paper, we will define a few more concepts common
to the study of geometry.
Definition 8. A subset X ⊆ Rn is said to be convex if x, y ∈ X implies that λx + (1 − λ)y ∈ X for all λ ∈ [0, 1].
To aid in the intuitive understanding of convexity, note that this simply means the line segment connecting x to y
also lies entirely within X. X is said to be symmetric if x ∈ X implies that −x ∈ X. This simply means that X
is invariant under reflection through the origin. Furthermore, X is called bounded if there exists an n-dimensional
sphere S (of finite radius) so that X ⊂ S. Since convexity is, in our experience, the most difficult to grasp, Fig. 1
provides a few illustrations to aid in understanding it.
FIG. 1. Examples of a Convex and a Non-Convex Set
b
x
b
Convex
y
Non-convex
Theorem 5 (Minkowski, 1896). Let D be the fundamental domain of an n-dimensional lattice L. Further, let X be
a bounded, symmetric, convex subset of L. If
V (X) > 2n V (D)
then X contains at least one non-zero lattice vector (i.e. element of L).
Proof. We begin by choosing a basis of L, say v1 , · · · , vn and consider the n-dimensional lattice 2L generated by
2v1 , · · · , 2vn and its fundamental domain 2D. We can see that
v(2D) = 2n v(D)
since we have simply doubled each of the dimensions, resulting in an overall change of 2n – one factor of 2 for each
dimension (just think of simple case of the volume of a unit sphere and what happens when you double the length of
each side). Notice, however, that this doubling does not affect the isomorphism with the n-dimensional torus, i.e. we
have
Rn /2L ∼
= Tn ,
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which, when coupled with the above discussion of the volume of 2D, yields
v(Tn ) = v(2D) = 2n v(D).
But this means that
v(φ(X)) ≤ v(Tn ) = 2n v(D) < v(X),
so by Theorem 4 we have that φ|X is not one-to-one. This furnishes some distinct x1 , x2 ∈ X so that φ(x1 ) = φ(x2 ),
which is equivalent to x1 − x2 ∈ 2L. Another way of thinking of this condition is to notice that
1
(x1 − x2 ) ∈ L.
2
Thus convexity and symmetry of X ensure that
1
1
1
(x1 − x2 ) = x1 + 1 −
(−x2 ) ∈ X.
2
2
2
1
This mean that (x1 − x2 ) is a lattice vector in X which is nonzero since x1 and x2 are distinct; thus we have found
2
the element we were looking for.
This result may seem (much like the theorem on the isomorphism between the quotient group Rn /L and the torus
T ) to be inconsequential. It is this theorem, however, upon which nearly all aspects of lattice geometry lie, as well
as a vast wealth of results in algebraic number theory. If one wishes to get a grasp on the geometry of the internal
structure of crystals, then understanding this theorem provides a crucial stepping stone in that direction.
n
[1] D.S. Dummit and R.M. Foote. Abstract Algebra. Wiley, 3rd edition, 2003.
[2] A. Frölich and M.J. Taylor. Algebraic Number Theory. Cambridge Studies in Advanced Mathematics, 27. Cambridge
University Press, 1993.
[3] T. Hungerford. Algebra. Graduate Texts in Mathematics, 73. Springer, 1980.
[4] H.W. Lenstra. Lattices, 2008. http://www.math.leidenuniv.nl/~ psh/ANTproc/06hwl.pdf.
[5] J.S. Milne. Algebraic number theory (v3.03). www.jmilne.org/math/, 2011.
[6] R. Virk. The Geometry of Numbers. http://www.math.ucdavis.edu/~ virk/notes/pre08/pdf/geometryofnumbers.pdf.