Introduction to Coxeter Groups

Transcription

1 OSU April 25, mdavis/

2 1 Geometric reflection groups Some history Properties 2

3 Some history 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 D m denote the dihedral gp corresponding to m. For m, D m can be represented as the subgp of O(2) which is generated by reflections across lines L, L, making an angle of π/m. π/m r L r L

4 Some history Properties - In 1852 Möbius 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 was a spherical triangle with angles π p, π q, π r, with p, q, r integers 2. - Since the sum of the angles is > π, we have 1 p + 1 q + 1 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 E 2 or H 2 generated by reflections across the edges of triangles with angles integral submultiples of π. Poincaré and Klein: similarly for polygons in H 2.

5 Some history Properties In 2 nd half of the 19 th century work began on finite reflection gps on S n, n > 2, generalizing Möbius results for n = 2. It developed along two lines. - Around 1850, Schläfli classified regular polytopes in R n+1, n > 2. The symmetry group of such a polytope was a finite gp generated by reflections and as in Möbius case, the projection of a fundamental domain to S n 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 S n or E n.

6 Some history Properties Let K be a fundamental polytope for a geometric reflection gp. For S n, K is a simplex. For E n, K is a product of simplices. For H n there are other possibilities, eg, a right-angled pentagon in H 2 or a right-angled dodecahedron in H 3.

7 Some history Properties Conversely, given a convex polytope K in S n, E n or H n 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), and for s t, the dihedral angle along that face is π/m(s, t). Moreover, S (st) m(s,t), where (s, t) S S is a presentation for W.

8 Some history 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) 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.

9 Some history Properties Spherical Diagrams Euclidean Diagrams A n A n B n 4 B n 4 D n C n 4 4 I 2(p) p D n H3 H A 1 B 2 G 2 ω F 4 4 F 4 4 E 6 E 6 E 7 E 7 E 8 E 8

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

11 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: S (st) m(s,t), where (s, t) S S. 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!

12 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: S (st) m(s,t), where (s, t) S S. where m(s, t) is a Coxeter matrix.

13 Terminology Given a subset T S, put W T := T (called a special subgp). T is spherical if W T 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.)

14 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 ).

15 Question Can every Coxeter system be geometrized? Answer Yes. In fact, there are two different answers.

16 The basic construction A mirror structure on a space X is a family of closed subspaces {X s } s S. For x X, put S(x) = {s S x X s }. Define U(W, X) := (W X)/, where is the equivalence relation: (w, x) (w, x ) x = x and w 1 w W S(x) (the subgp generated by S(x)). U(W, X) is formed by gluing together copies of X (the chambers). W U(W, X).

17 First answer: the Tits representation Tits showed a linear action of W on R S generated by (not necessarily orthogonal) reflections across the faces of the standard simplicial cone C R S giving a faithful linear representation W GL(R S ). Moreover, WC (= w W wc) is a convex cone and if I denotes the interior of the cone, then I = U(W, C f ), where C f is the complement of the nonspherical faces of C and U(W, C f ) is defined as before, ie, U(W, C f ) = (W C f )/.

18 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 C f is not compact).

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

20 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).

21 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: K s := S {s}. Σ := U(W, K ).

22 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, C f ) = I. Σ is the cocompact core of I.

23 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.

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

25 Coxeter zonotopes Suppose W T is finite reflection gp on R T. Choose a point x in the interior of fundamental simplicial cone and let P T be convex hull of W T x. s st t s 1 s t sts= tst t s ts The 1-skeleton of P T is Cay(W T, T ). t

26 Filling in Cay(W, S) When W T = (Z/2) n, then P T is an n-cube. There is a cell structure on Σ with {cells} = W S. This follows from fact that poset of cells in P T is = W T S T. The cells of Σ are defined as follows: the geometric realization of subposet of cosets ww T is = barycentric subdiv of P T.

27 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).

28 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 W T ). - 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 X n = E n or H n, then K can be identified with the fundamental polytope, Σ with X n and the cell structure is dual to the tessellation of Σ by translates of K.