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• COXETER GROUPS

Gert Heckman

[email protected]

January 26, 2017

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• 2

• Contents

Preface 4

1 Regular Polytopes 7

1.1 Convex Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2 Examples of Regular Polytopes . . . . . . . . . . . . . . . . . 121.3 Classification of Regular Polytopes . . . . . . . . . . . . . . . 16

2 Finite Reflection Groups 21

2.1 Normalized Root Systems . . . . . . . . . . . . . . . . . . . . 212.2 The Dihedral Normalized Root System . . . . . . . . . . . . . 242.3 The Basis of Simple Roots . . . . . . . . . . . . . . . . . . . . 252.4 The Classification of Elliptic Coxeter Diagrams . . . . . . . . 272.5 The Coxeter Element . . . . . . . . . . . . . . . . . . . . . . . 352.6 A Dihedral Subgroup of W . . . . . . . . . . . . . . . . . . . 392.7 Integral Root Systems . . . . . . . . . . . . . . . . . . . . . . 422.8 The Poincare Dodecahedral Space . . . . . . . . . . . . . . . 46

3 Invariant Theory for Reflection Groups 53

3.1 Polynomial Invariant Theory . . . . . . . . . . . . . . . . . . 533.2 The Chevalley Theorem . . . . . . . . . . . . . . . . . . . . . 563.3 Exponential Invariant Theory . . . . . . . . . . . . . . . . . . 60

4 Coxeter Groups 65

4.1 Generators and Relations . . . . . . . . . . . . . . . . . . . . 654.2 The Tits Theorem . . . . . . . . . . . . . . . . . . . . . . . . 694.3 The Dual Geometric Representation . . . . . . . . . . . . . . 744.4 The Classification of Some Coxeter Diagrams . . . . . . . . . 774.5 Affine Reflection Groups . . . . . . . . . . . . . . . . . . . . . 864.6 Crystallography . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5 Hyperbolic Reflection Groups 97

5.1 Hyperbolic Space . . . . . . . . . . . . . . . . . . . . . . . . . 975.2 Hyperbolic Coxeter Groups . . . . . . . . . . . . . . . . . . . 1005.3 Examples of Hyperbolic Coxeter Diagrams . . . . . . . . . . . 1085.4 Hyperbolic reflection groups . . . . . . . . . . . . . . . . . . . 1145.5 Lorentzian Lattices . . . . . . . . . . . . . . . . . . . . . . . . 116

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• 6 The Leech Lattice 125

6.1 Modular Forms . . . . . . . . . . . . . . . . . . . . . . . . . . 1256.2 A Theorem of Venkov . . . . . . . . . . . . . . . . . . . . . . 1296.3 The Classification of Niemeier Lattices . . . . . . . . . . . . . 1326.4 The Existence of the Leech Lattice . . . . . . . . . . . . . . . 1336.5 A Theorem of Conway . . . . . . . . . . . . . . . . . . . . . . 1356.6 The Covering Radius of . . . . . . . . . . . . . . . . . . . . 1376.7 Uniqueness of the Leech Lattice . . . . . . . . . . . . . . . . . 140

4

• Preface

Finite reflection groups are a central subject in mathematics with a longand rich history. The group of symmetries of a regular m-gon in the plane,that is the convex hull in the complex plane of the mth roots of unity, is thedihedral group Dm of order 2m, which is the simplest example of a reflectiongroup. Similarly the groups of symmetries of a tetrahedron, octahedron andicosahedron in Euclidean three space are isomorphic to S4, S4 C2 andA5 C2 respectively, and are again finite reflection groups. More generally,the symmetry groups of a regular polytope in Euclidean space of dimensionn are examples of finite reflection groups. Such polytopes were classifiedby Ludwig Schlafli in 1852 [50]. In dimension n 5 there are just threeof them, the simplex of dimension n and the dual pair of hyperoctahedronand hypercube. Their symmetry groups are the symmetric group Sn+1 andthe hyperoctahedral group Hn respectively. In dimension n = 4 there arethree additional regular polytopes, and all their symmetry groups are finitereflection groups [23], [4].

Reflection groups were studied in a systematic way by Donald Coxeter(1907-2003) in the nineteen thirties. Not just the finite reflection groups ina Euclidean vector space, but also infinite reflection groups in a Euclideanaffine space or a Lobachevsky space. If you like to see a glimpse of Cox-eter then you can look up Coxeter discusses the math behind Escherscircle limit on YouTube. Their theory was further developped by JacquesTits, leading to the concept of Coxeter groups [9] and giving a basis forTits geometry. Hyperbolic reflection groups were studied extensively byErnest Vinberg and his Russian colleagues [61]. The delightful biographyby Shiobhan Roberts about the life and work of Donald Coxeter is highlyrecommended [49].

The true reason of their importance in mathematics is the fundamentalrole played by reflection groups in the theory of simple algebraic groups andsimple Lie algebras, as established in the late 19th and early 20th centuryby Wilhelm Killing, Elie Cartan and Hermann Weyl [33], [13], [66]. Both intheir classification and for the formulation of the the basic results reflectiongroups play a pivitol role. But there are many more subjects, for examplethe study of surface singularities, the theory of hypergeometric functions,the theory of integrable models or in various parts of algebraic geometry,where reflection groups again play a crucial role. Reflection groups are a trueubiquity in modern mathematics in a remarkable and almost unreasonablevariety of different ways.

There are several good text books, in which the material of Chapters 2,

5

• 3, 4 and 5 is covered. The Bourbaki text Chapitres 4, 5 et 6 of Groupes etAlgebres de Lie from 1968 is a jewel (and by far the most successful book inthe whole Bourbaki series, as can for example be seen from a citation countof N. Bourbaki in MathSciNet). Besides the text of this Bourbaki volumethe given Tables are also quite useful. Another well written book on thesubject is the text by James Humphreys on Reflection and Coxeter groupsfrom 1990, but his exposition on hyperbolic reflection groups is somewhatlimited.

6

• 1 Regular Polytopes

1.1 Convex Sets

Let V be a Euclidean vector space of dimension n with scalar product (, ).We shall denote scalars by Roman letters x, y, and vectors in V by Greekletters , , .

Definition 1.1. A subset A V is called affine if for all , A and allx R we have (1 x) + x A.

Obviously the intersection of a collection of affine sets in V is againaffine. For X V a subset of V we denote by aff(X) the intersection ofall affine subsets in V containing X. Clearly the affine hull aff(X) is thesmallest affine subset of V containing X.

Lemma 1.2. If A V is a nonempty affine set and 0, 1, , m Aand x0, x1, , xm R with

xi = 1 then

xii A (with sum over0 i m).

Proof. This follows easily by induction of the natural number m. The casem = 1 is clear from the definition of affine set. Now suppose m 2 and sayxm 6= 1. Then we can write

m

0

xii = (1 xm)(m1

0

yjj) + xmm

with yj = xj/(1 xm) and

yj = (

xj)/(1 xm) = 1 (with sum over0 j m 1). By the induction hypothesis yjj A and hence also

xii A.

Corollary 1.3. If A is a nonempty affine set in V and 0 A then A isjust the translate 0 + L over 0 of a linear subspace L V .

Proof. We take A = 0 + L as definition of the subset L of V . If x, y Rand , L then 0 + x + y = (1 x y)0 + x(0 + ) + y(0 + ) A.Hence L is a linear subspace of V .

Hence we can speak of the dimension of an affine subset A of V : dimA =dimL. If dimA = 0, 1, 2,m 1 then A is called a point, a line, a plane or ahyperplane respectively.

Definition 1.4. A subset C V is called convex if for all , C and all0 x 1 we have (1 x) + x C.

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• The intersection of a collection of convex sets in V is again convex. ForX V the convex hull conv(X) is defined as the intersection of all convexsets in V containing X. Clearly conv(X) is the smallest convex set in Vcontaining X.

Lemma 1.5. If C V is a nonempty convex set and 0, 1, , m Cand x0, x1, , xm 0 with

xi = 1 then

xii C (with sum over0 i m).

Proof. The proof is the same as that for Lemma 1.2.

Lemma 1.6. If C V is a nonempty convex set then the relative interiorrelint(C) of C in the affine hull aff(C) is nonempty.

Proof. If aff(C) has dimension m then we can choose (m+1) distinct points0, 1, , m C with aff(C) = aff({0, 1, , m}). Now the msimplexconv({0, 1, , m}) is contained in C and has nonempty relative interiorin aff(C).

Hence we can define the dimension of a nonempty convex set C V bydimC = dimaff(C). Throughout the remaining part of this section let C bea nonempty closed convex set in V . Suppose V is a point. Because C isclosed there is a point C at minimal distance from : | | | |for all C. Because C is convex this point C is unique and we write = p() = pC().

Definition 1.7. The map p = pC : V C is called the metric projectionof V on C.

Definition 1.8. For C the set

N() = NC() = { V ; (, ) 0 C}

is called the outer normal cone of C at .

Lemma 1.9. For C the set N() is a closed convex cone.

Proof. Clearly N() is closed as intersection of closed sets. For 1, 2 N()and x1, x2 0 we have

(x11 + x22, ) = x1(1, ) + x2(2, ) 0

for all C, and therefore x11 + x22 N().

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• Proposition 1.10. For V and C we have p() = if and only if( ) N().Proof. For V and C we have p() = if and only if

( , ) ( (1 x) x, (1 x) x) 0for all C and all 0 x 1. In turn this is equivalent to

(2( ) + x( ), x( )) 0for all C and all 0 x 1, or equivalently

(2( ) + x( ), ) 0for all C and all 0 x 1, or equivalently

(2( ), ) 0for all C. This amounts to ( ) N().

Theorem 1.11. The metric projection p : V C is a contraction, meaning|p() p()| | |

for all

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