Introduction to Coxeter Groups - Department of Mathematics · 2011-04-25 · Mike Davis...

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Geometric reflection groups Abstract reflection groups Introduction to Coxeter Groups Mike Davis OSU April 25, 2011 http://www.math.ohio-state.edu/mdavis/ Mike Davis Introduction to Coxeter Groups

Transcript of Introduction to Coxeter Groups - Department of Mathematics · 2011-04-25 · Mike Davis...

Geometric reflection groupsAbstract reflection groups

Introduction to Coxeter Groups

Mike Davis

OSU

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

Mike Davis Introduction to Coxeter Groups

Geometric reflection groupsAbstract reflection groups

1 Geometric reflection groupsSome historyProperties

2 Abstract reflection groupsCoxeter systemsThe Tits representationThe cell complex Σ

Mike Davis Introduction to Coxeter Groups

Geometric reflection groupsAbstract reflection groups

Some historyProperties

Dihedral groupsA dihedral gp is any gp which is generated by 2 involutions, callthem s, t . It is determined up to isomorphism by the order m ofst (m is an integer ≥ 2 or the symbol∞). Let Dm denote thedihedral gp corresponding to m.

For m 6=∞, Dm can berepresented as the subgp ofO(2) which is generated byreflections across lines L, L′,making an angle of π/m.

rL

rL´

π/m

Mike Davis Introduction to Coxeter Groups

Geometric reflection groupsAbstract reflection groups

Some historyProperties

- 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-spherewas a spherical triangle with angles π

p , πq , πr , with p, q, rintegers ≥ 2.

- Since the sum of the angles is > π, we have 1p + 1

q + 1r > 1.

- For p ≥ q ≥ r , the only possibilities are: (p,2,2) for anyp ≥ 2 and (p,3,2) with p = 3, 4 or 5. The last three casesare the symmetry groups of the Platonic solids.

- Later work by Riemann and Schwarz showed there werediscrete gps of isometries of E2 or H2 generated byreflections across the edges of triangles with anglesintegral submultiples of π. Poincare and Klein: similarly forpolygons in H2.

Mike Davis Introduction to Coxeter Groups

Geometric reflection groupsAbstract reflection groups

Some historyProperties

In 2nd half of the 19th century work began on finite reflectiongps on Sn, n > 2, generalizing Mobius’ results for n = 2. Itdeveloped along two lines.

- Around 1850, Schlafli classified regular polytopes in Rn+1,n > 2. The symmetry group of such a polytope was a finitegp generated by reflections and as in Mobius’ case, theprojection of a fundamental domain to Sn was a sphericalsimplex with dihedral angles integral submultiples of π.

- Around 1890, Killing and E. Cartan classified complexsemisimple Lie algebras in terms of their root systems. In1925, Weyl showed the symmetry gp of such a root systemwas a finite reflection gp.

- These two lines were united by Coxeter in the 1930’s. Heclassified discrete groups reflection gps on Sn or En.

Mike Davis Introduction to Coxeter Groups

Geometric reflection groupsAbstract reflection groups

Some historyProperties

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. ForHn there are other possibilities, eg, a right-angled pentagon inH2 or a right-angled dodecahedron in H3.

Mike Davis Introduction to Coxeter Groups

Geometric reflection groupsAbstract reflection groups

Some historyProperties

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

〈S | (st)m(s,t), where (s, t) ∈ S × S〉

is a presentation for W .

Mike Davis Introduction to Coxeter Groups

Geometric reflection groupsAbstract reflection groups

Some historyProperties

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. Labelthe edge by m(s, t) if this is > 3 or =∞ and leave it unlabeled ifit is = 3. (W ,S) is irreducible if Γ is connected. (Thecomponents of Γ give the irreducible factors of W .)The next slide shows Coxeter’s classification of irreduciblespherical and cocompact Euclidean reflection gps.

Mike Davis Introduction to Coxeter Groups

Geometric reflection groupsAbstract reflection groups

Some historyProperties

Spherical Diagrams Euclidean Diagrams

I (p) p2

H 53

H 54

F 44

ωA1∼

4 4B 2∼

6G2

4F4

E6

E7

E8

E6

E7

E8

An

B 4n

Dn

nA∼

4B n∼

4 4Cn∼

Dn∼

Mike Davis Introduction to Coxeter Groups

Geometric reflection groupsAbstract reflection groups

Coxeter systemsThe Tits representationThe cell complex Σ

QuestionGiven 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

Geometric reflection groupsAbstract reflection groups

Coxeter systemsThe Tits representationThe cell complex Σ

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

〈S | (st)m(s,t), where (s, t) ∈ S × S〉.

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

These two answers are equivalent!

Mike Davis Introduction to Coxeter Groups

Geometric reflection groupsAbstract reflection groups

Coxeter systemsThe Tits representationThe cell complex Σ

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

〈S | (st)m(s,t), where (s, t) ∈ S × S〉.

where m(s, t) is a Coxeter matrix.

Mike Davis Introduction to Coxeter Groups

Geometric reflection groupsAbstract reflection groups

Coxeter systemsThe Tits representationThe cell complex Σ

Terminology

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

DefinitionThe 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

Geometric reflection groupsAbstract reflection groups

Coxeter systemsThe Tits representationThe cell complex Σ

Definition (RACS)

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

Suppose Γ is a graph with Vert Γ = S. Define a Coxeter matrixwith off-diagonal elements given by

m(s, t) =

{2, if {s, t} ∈ Edge Γ

∞, otherwise.

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

Mike Davis Introduction to Coxeter Groups

Geometric reflection groupsAbstract reflection groups

Coxeter systemsThe Tits representationThe cell complex Σ

QuestionCan every Coxeter system be geometrized?

AnswerYes. In fact, there are two different answers.

Mike Davis Introduction to Coxeter Groups

Geometric reflection groupsAbstract reflection groups

Coxeter systemsThe Tits representationThe cell complex Σ

The basic constructionA 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 ′, x ′) ⇐⇒x = x ′ and w−1w ′ ∈WS(x) (the subgp generated by S(x)).U(W ,X ) is formed by gluing together copies of X (thechambers). W y U(W ,X ).

Mike Davis Introduction to Coxeter Groups

Geometric reflection groupsAbstract reflection groups

Coxeter systemsThe Tits representationThe cell complex Σ

First answer: the Tits representation

Tits showed ∃ a linear action of W on RS generated by (notnecessarily orthogonal) reflections across the faces of thestandard simplicial cone C ⊂ RS giving a faithful linearrepresentation W ↪→ GL(RS).

Moreover, WC (=⋃

w∈W wC) is a convex cone and if Idenotes the interior of the cone, thenI = U(W ,Cf ), where Cf is the complement of thenonspherical faces of C and U(W ,Cf ) is defined as before,ie, U(W ,Cf ) = (W × Cf )/ ∼.

Mike Davis Introduction to Coxeter Groups

Geometric reflection groupsAbstract reflection groups

Coxeter systemsThe Tits representationThe cell complex Σ

One consequenceW is virtually torsion-free. (This is true for any finitely generatedlinear gp.)

AdvantagesI 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

Geometric reflection groupsAbstract reflection groups

Coxeter systemsThe Tits representationThe cell complex Σ

RemarkWe also have a rep W → PGL(Rn). So, W y PI, theimage of I in projective space. When W is infinite andirreducible, this is a proper convex subset of RPn−1.Vinberg showed one can get linear representations acrossthe faces of polyhedral cones of lower dimension.

Mike Davis Introduction to Coxeter Groups

Geometric reflection groupsAbstract reflection groups

Coxeter systemsThe Tits representationThe cell complex Σ

The second answer is to construct of contractible cell cx Σ onwhich W acts properly and cocompactly as a gp generated byreflections.

There are two dual constructions of Σ.

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

Mike Davis Introduction to Coxeter Groups

Geometric reflection groupsAbstract reflection groups

Coxeter systemsThe Tits representationThe cell complex Σ

Geometric realization of a poset

Given a poset P, let Flag(P) denote the abstract simplicialcomplex with vertex set P and with simplices all finite, totallyordered subsets of P. The geometric realization of Flag(P) isdenoted |P|.

First construction of Σ

Recall S is the poset of spherical subsets of S. Thefundamental chamber K is defined by K := |S|. (K is thecone on the barycentric subdivision of L.)Mirror structure: Ks := |S≥{s}|.Σ := U(W ,K ).

Mike Davis Introduction to Coxeter Groups

Geometric reflection groupsAbstract reflection groups

Coxeter systemsThe Tits representationThe cell complex Σ

Relationship with Tits representationSuppose W is infinite. Then K is subcx of b∆, thebarycentric subdivision of the simplex ∆ ⊂ C.Consider the vertices which are barycenters of sphericalfaces. They span a subcx of b∆. This subcx is K . It is asubset 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

Geometric reflection groupsAbstract reflection groups

Coxeter systemsThe Tits representationThe 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

Geometric reflection groupsAbstract reflection groups

Coxeter systemsThe Tits representationThe cell complex Σ

Second constructionLet WS denote the disjoint union of all spherical cosets(partially ordered by inclusion):

WS :=∐T∈S

W/WT and Σ := |WS|.

Mike Davis Introduction to Coxeter Groups

Geometric reflection groupsAbstract reflection groups

Coxeter systemsThe Tits representationThe cell complex Σ

Coxeter zonotopes

Suppose WT is finite reflection gp on RT . Choose a point x inthe interior of fundamental simplicial cone and let PT be convexhull of WT x .

t

s

1

t

s

s

s

t

t

st

sts= tst

ts

The 1-skeleton of PT is Cay(WT ,T ).

Mike Davis Introduction to Coxeter Groups

Geometric reflection groupsAbstract reflection groups

Coxeter systemsThe Tits representationThe cell complex Σ

Filling in Cay(W ,S)

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

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

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

Mike Davis Introduction to Coxeter Groups

Geometric reflection groupsAbstract reflection groups

Coxeter systemsThe Tits representationThe 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

Geometric reflection groupsAbstract reflection groups

Coxeter systemsThe Tits representationThe 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 reflectiongp on Xn = En or Hn, then Kcan be identified with thefundamental polytope, Σ withXn and the cell structure isdual to the tessellation of Σby translates of K .

Mike Davis Introduction to Coxeter Groups