Introduction to Coxeter Groups

Geometric reﬂection groups Abstract reﬂection groups

Mike Davis

OSU

April 25, 2011 http://www.math.ohio-state.edu/∼mdavis/

Mike Davis Introduction to Coxeter Groups Geometric reﬂection groups Abstract reﬂection groups

1 Geometric reﬂection groups Some history Properties

2 Abstract reﬂection groups Coxeter systems The Tits representation The cell complex Σ

Mike Davis Introduction to Coxeter Groups Geometric reﬂection groups Some history Abstract reﬂection groups Properties

Dihedral groups A dihedral gp is any gp which is generated by 2 involutions, call them s, t. It is determined up to isomorphism by the order m of st (m is an integer ≥ 2 or the symbol ∞). Let Dm denote the dihedral gp corresponding to m.

r For m 6= ∞, Dm can be L represented as the subgp of

π/m r O(2) which is generated by L´ reﬂections across lines L, L0, making an angle of π/m.

Mike Davis Introduction to Coxeter Groups Geometric reﬂection groups Some history Abstract reﬂection groups Properties

- In 1852 Mobius¨ determined the finite subgroups of O(3) generated by isometric reflections on the 2-sphere. - The fundamental domain for such a group on the 2-sphere π π π was a spherical triangle with angles p , q , r , with p, q, r integers ≥ 2. 1 1 1 - Since the sum of the angles is > π, we have p + q + r > 1. - For p ≥ q ≥ r, the only possibilities are: (p, 2, 2) for any p ≥ 2 and (p, 3, 2) with p = 3, 4 or 5. The last three cases are the symmetry groups of the Platonic solids. - Later work by Riemann and Schwarz showed there were discrete gps of isometries of E2 or H2 generated by reflections across the edges of triangles with angles integral submultiples of π. Poincare´ and Klein: similarly for polygons in H2.

Mike Davis Introduction to Coxeter Groups Geometric reﬂection groups Some history Abstract reﬂection groups Properties

In 2nd half of the 19th century work began on finite reflection gps on Sn, n > 2, generalizing Mobius’¨ results for n = 2. It developed along two lines. - Around 1850, Schlafli¨ classified regular polytopes in Rn+1, n > 2. The symmetry group of such a polytope was a finite gp generated by reflections and as in Mobius’¨ case, the projection of a fundamental domain to Sn was a spherical simplex with dihedral angles integral submultiples of π. - Around 1890, Killing and E. Cartan classified complex semisimple Lie algebras in terms of their root systems. In 1925, Weyl showed the symmetry gp of such a root system was a finite reflection gp. - These two lines were united by Coxeter in the 1930’s. He classified discrete groups reflection gps on Sn or En.

Mike Davis Introduction to Coxeter Groups Geometric reﬂection groups Some history Abstract reﬂection groups Properties

Let K be a fundamental polytope for a geometric reﬂection gp. For Sn, K is a simplex. For En, K is a product of simplices. For Hn there are other possibilities, eg, a right-angled pentagon in H2 or a right-angled dodecahedron in H3.

Mike Davis Introduction to Coxeter Groups Geometric reﬂection groups Some history Abstract reﬂection groups Properties

Conversely, given a convex polytope K in Sn, En or Hn st all dihedral angles have form π/integer, there is a discrete gp W generated by isometric reﬂections across the codim 1 faces of K . Let S be the set of reﬂections across the codim 1 faces of K . For s, t ∈ S, let m(s, t) be the order of st. Then S generates W , the faces corresponding to s and t intersect in a codim 2 face iff m(s, t) 6= ∞, and for s 6= t, the dihedral angle along that face is π/m(s, t). Moreover,

hS | (st)m(s,t), where (s, t) ∈ S × Si is a presentation for W .

Mike Davis Introduction to Coxeter Groups Geometric reﬂection groups Some history Abstract reﬂection groups Properties

Coxeter diagrams Associated to (W , S), there is a labeled graph Γ called its “Coxeter diagram.” Vert(Γ) := S. Connect distinct elements s, t by an edge iff m(s, t) 6= 2. Label the edge by m(s, t) if this is > 3 or = ∞ and leave it unlabeled if it is = 3. (W , S) is irreducible if Γ is connected. (The components of Γ give the irreducible factors of W .) The next slide shows Coxeter’s classiﬁcation of irreducible spherical and cocompact Euclidean reﬂection gps.

Mike Davis Introduction to Coxeter Groups Geometric reﬂection groups Some history Abstract reﬂection groups Properties

Spherical Diagrams Euclidean Diagrams

A n ∼ A n

4 B n ∼ 4 B n

∼ 4 4 D n C n ∼ p D n I2 (p)

∼ ω 5 A 1 H3 ∼ 4 4 B 2 5 H 4 6 G∼ 2 4 F 4 4 F∼ 4

E 6 E∼ 6

E 7 E∼ 7

E E∼ 8 8

Mike Davis Introduction to Coxeter Groups Coxeter systems Geometric reﬂection groups The Tits representation Abstract reﬂection groups The cell complex Σ

Question Given a gp W and a set S of involutions which generates it, when should (W , S) be called an “abstract reﬂection gp”?

Mike Davis Introduction to Coxeter Groups Coxeter systems Geometric reﬂection groups The Tits representation Abstract reﬂection groups The cell complex Σ

Two answers Let Cay(W , S) be the Cayley graph (ie, its vertex set is W and {w, v} spans an edge iff v = ws for some s ∈ S). First answer: for each s ∈ S, the ﬁxed set of s separates Cay(W , S). Second answer: W has a presentation of the form:

hS | (st)m(s,t), where (s, t) ∈ S × Si.

Here m(s, t) is any S × S symmetric matrix with 1s on the diagonal and off-diagonal elements integers ≥ 2 or the symbol ∞.

These two answers are equivalent!

Mike Davis Introduction to Coxeter Groups Coxeter systems Geometric reﬂection groups The Tits representation Abstract reﬂection groups The cell complex Σ

If either answer holds, (W , S) is a Coxeter system and W a Coxeter group. The second answer is usually taken as the ofﬁcial deﬁnition: W has a presentation of the form:

hS | (st)m(s,t), where (s, t) ∈ S × Si. where m(s, t) is a Coxeter matrix.

Mike Davis Introduction to Coxeter Groups Coxeter systems Geometric reﬂection groups The Tits representation Abstract reﬂection groups The cell complex Σ

Terminology

Given a subset T ⊂ S, put WT := hT i (called a special subgp). T is spherical if WT is ﬁnite. Let S denote the poset of spherical subsets of S. (N.B. ∅ ∈ S.)

Deﬁnition The nerve of (W , S) is the simplicial complex L (= L(W , S)) with Vert(L) := S and with T ⊂ S a simplex iff T is spherical. (So, S(L) = S.)

Mike Davis Introduction to Coxeter Groups Coxeter systems Geometric reﬂection groups The Tits representation Abstract reﬂection groups The cell complex Σ

Deﬁnition (RACS) A Coxeter system (W , S) is right-angled (a RACS) if all off-diagonal m(s, t) are = 2 or ∞.

Suppose Γ is a graph with Vert Γ = S. Deﬁne a Coxeter matrix with off-diagonal elements given by ( 2, if {s, t} ∈ Edge Γ m(s, t) = ∞, otherwise.

A subset T ⊂ S is spherical iff it spans a complete subgraph of Γ. So, the nerve L is the associated ﬂag complex (= “clique complex”).

Mike Davis Introduction to Coxeter Groups Coxeter systems Geometric reﬂection groups The Tits representation Abstract reﬂection groups The cell complex Σ

Question Can every Coxeter system be geometrized?

Answer Yes. In fact, there are two different answers.

Mike Davis Introduction to Coxeter Groups Coxeter systems Geometric reﬂection groups The Tits representation Abstract reﬂection groups The cell complex Σ

The basic construction A mirror structure on a space X is a family of closed subspaces {Xs}s∈S. For x ∈ X, put S(x) = {s ∈ S | x ∈ Xs} . Deﬁne

U(W , X) := (W × X)/ ∼, where ∼ is the equivalence relation: (w, x) ∼ (w 0, x0) ⇐⇒ 0 −1 0 x = x and w w ∈ WS(x) (the subgp generated by S(x)). U(W , X) is formed by gluing together copies of X (the chambers). W y U(W , X).

Mike Davis Introduction to Coxeter Groups Coxeter systems Geometric reﬂection groups The Tits representation Abstract reﬂection groups The cell complex Σ First answer: the Tits representation

Tits showed ∃ a linear action of W on RS generated by (not necessarily orthogonal) reﬂections across the faces of the standard simplicial cone C ⊂ RS giving a faithful linear representation W ,→ GL(RS). S Moreover, WC (= w∈W wC) is a convex cone and if I denotes the interior of the cone, then I = U(W , Cf ), where Cf is the complement of the nonspherical faces of C and U(W , Cf ) is deﬁned as before, ie, U(W , Cf ) = (W × Cf )/ ∼.

Mike Davis Introduction to Coxeter Groups Coxeter systems Geometric reﬂection groups The Tits representation Abstract reﬂection groups The cell complex Σ

One consequence W is virtually torsion-free. (This is true for any ﬁnitely generated linear gp.)

Advantages I is contractible (since it is convex) and W acts properly (ie, with ﬁnite stabilizers) on it.

Disadvantage The action is not cocompact (since Cf is not compact).

Mike Davis Introduction to Coxeter Groups Coxeter systems Geometric reﬂection groups The Tits representation Abstract reﬂection groups The cell complex Σ

Remark We also have a rep W → PGL(Rn). So, W y PI, the image of I in projective space. When W is inﬁnite and irreducible, this is a proper convex subset of RPn−1. Vinberg showed one can get linear representations across the faces of polyhedral cones of lower dimension.

Mike Davis Introduction to Coxeter Groups Coxeter systems Geometric reﬂection groups The Tits representation Abstract reﬂection groups The cell complex Σ

The second answer is to construct of contractible cell cx Σ on which W acts properly and cocompactly as a gp generated by reﬂections.

There are two dual constructions of Σ.

Build the correct fundamental chamber K , then apply the basic construction, U(W , K ). Fill in the Cayley graph of (W , S).

Mike Davis Introduction to Coxeter Groups Coxeter systems Geometric reﬂection groups The Tits representation Abstract reﬂection groups The cell complex Σ

Geometric realization of a poset Given a poset P, let Flag(P) denote the abstract simplicial complex with vertex set P and with simplices all ﬁnite, totally ordered subsets of P. The geometric realization of Flag(P) is denoted |P|.

First construction of Σ Recall S is the poset of spherical subsets of S. The fundamental chamber K is deﬁned by K := |S|.(K is the cone on the barycentric subdivision of L.)

Mirror structure: Ks := |S≥{s}|. Σ := U(W , K ).

Mike Davis Introduction to Coxeter Groups Coxeter systems Geometric reﬂection groups The Tits representation Abstract reﬂection groups The cell complex Σ

Relationship with Tits representation Suppose W is inﬁnite. Then K is subcx of b∆, the barycentric subdivision of the simplex ∆ ⊂ C. Consider the vertices which are barycenters of spherical faces. They span a subcx of b∆. This subcx is K . It is a subset of ∆f . So, Σ = U(W , K ) ⊂ U(W , ∆f ) ⊂ U(W , Cf ) = I. Σ is the “cocompact core” of I.

Mike Davis Introduction to Coxeter Groups Coxeter systems Geometric reﬂection groups The Tits representation Abstract reﬂection groups The cell complex Σ

Groups, Graphs and Trees An Introduction to the Geometry of Inﬁnite Groups

John Meier Jon McCammond

Σ is the cocompact core of the Tits cone I.

Mike Davis Introduction to Coxeter Groups Coxeter systems Geometric reﬂection groups The Tits representation Abstract reﬂection groups The cell complex Σ

Second construction Let W S denote the disjoint union of all spherical cosets (partially ordered by inclusion): a W S := W /WT and Σ := |W S|. T ∈S

Mike Davis Introduction to Coxeter Groups Coxeter systems Geometric reﬂection groups The Tits representation Abstract reﬂection groups The cell complex Σ

Coxeter zonotopes T Suppose WT is ﬁnite reﬂection gp on R . Choose a point x in the interior of fundamental simplicial cone and let PT be convex hull of WT x.

s t s st 1

s t

sts= tst t t s ts The 1-skeleton of PT is Cay(WT , T ).

Mike Davis Introduction to Coxeter Groups Coxeter systems Geometric reﬂection groups The Tits representation Abstract reﬂection groups The cell complex Σ Filling in Cay(W , S)

n When WT = (Z/2) , then PT is an n-cube.

There is a cell structure on Σ with {cells} = W S.

∼ This follows from fact that poset of cells in PT is = WT S≤T . The cells of Σ are deﬁned as follows: the geometric realization ∼ of subposet of cosets ≤ wWT is = barycentric subdiv of PT .

Mike Davis Introduction to Coxeter Groups Coxeter systems Geometric reﬂection groups The Tits representation Abstract reﬂection groups The cell complex Σ

Properties of this cell structure on Σ - W acts cellularly on Σ. - Σ has one W -orbit of cells for each spherical subset T ∈ S and dim(cell) = Card(T ). - The 0-skeleton of Σ is W - The 1-skeleton of Σ is Cay(W , S). - The 2-skeleton of Σ is the Cayley 2 complex of the presentation. - If W is right-angled, then each Coxeter cell is a cube. - Moussong: the induced piecewise Euclidean metric on Σ is CAT(0).

Mike Davis Introduction to Coxeter Groups Coxeter systems Geometric reﬂection groups The Tits representation Abstract reﬂection groups The cell complex Σ

More properties - Σ is contractible. (This follows from the fact it is CAT(0)). - The W -action is proper (by construction each isotropy subgp is conjugate to some ﬁnite WT ). - Σ/W = K , which is compact (so the action is cocompact) - If W is ﬁnite, then Σ is a Coxeter zonotope.

If W is a geometric reﬂection gp on Xn = En or Hn, then K can be identiﬁed with the fundamental polytope, Σ with Xn and the cell structure is dual to the tessellation of Σ by translates of K .

Mike Davis Introduction to Coxeter Groups