Coxeter Complexes
3.1 The Coxeter Complex
3
Definition 3.3.The elements of Σ are called simplices. The maximal sim- plices, which are the singletons {w}, are called chambers and are identified with the elements of W. The simplices of the form ws = {w, ws} (with w∈W ands∈S) are calledpanels. We setC:= 1 and call it thefundamen- tal chamber. Each panel wsis a face of exactly two chambers, w and ws, which are said to bes-adjacent.
Note that there is an action ofW onΣby left translation, and the action on the chambers is simply transitive. Note further that we can construct
“galleries” inΣin the way that is familiar from Chapter 1: Givenw∈W and a decompositionw=s1· · ·sl (withsi∈S), we have a sequence of chambers
Γ:C=C0, Cl, . . . , Cl=wC , (3.1) whereCi=s1· · ·siC, andCi−1 issi-adjacent toCi fori= 1, . . . , l.
At this point the reader may need to refer to Section A.1.3 in Appendix A for the terminology regarding chamber complexes and type functions. Let’s add one more bit of terminology:
Definition 3.4.A chamber complex is calledthin if every panel is a face of exactly two chambers.
Theorem 3.5.The poset Σ:=Σ(W, S)is a simplicial complex. Moreover, it is a thin chamber complex of rank equal to|S|, it is colorable, and the action of W on Σ is type-preserving.
Proof. To show thatΣ is simplicial, there are two things we must verify (see Definition A.1):
(a) Any two elementsA, B∈Σ have a greatest lower bound.
(b) For anyA∈Σ, the posetΣ≤A is a Boolean lattice.
For (a) we can use theW-action on Σ to reduce to the case that one of the two elements is a face of the fundamental chamber C, i.e., is a standard subgroup. What we must prove, then, is that a standard subgroupWJ and a standard cosetwWK (whereJ, K⊆S) have a least upper bound in the set of standard cosets, with respect to the ordering by inclusion. Now any standard coset containing the two given ones contains the identity and hence is a stan- dard subgroup. Moreover, it contains w and hence also WK = w−1(wWK).
So the upper bounds of our two standard cosets are the standard subgroups containingJ,K, andw. In view of Proposition 2.16, there is indeed a smallest upper bound, namely, the standard subgroupWL, where L:=J∪K∪S(w).
To prove (b), we may assume that A is the fundamental chamber C. In this case, Σ≤C is the set of standard subgroups of W (ordered by reverse inclusion). By Proposition 2.13 we have
Σ≤C∼= (subsets ofS)op∼= (subsets ofS),
3.1 The Coxeter Complex 117 where the second isomorphism is given byJ →SJforJ ⊆S. This proves (b) and completes the proof that Σ is simplicial. The proof also shows that all maximal simplices have the same rank, equal to|S|. And the discussion sur- rounding (3.1) above implies that any two of them can be connected by a gallery and that any panel is a face of exactly two chambers. So Σis a thin chamber complex.
Finally, we can define a W-invariant type function τ on Σ, with values
in S, by settingτ(wWJ) :=SJ.
We will continue to denote byτthe type function constructed in the proof.
For emphasis, we repeat the definition:
Definition 3.6.Σ(W, S) has acanonical type function with values inS, de- fined by
τ(wWJ) =SJ
forw∈W andJ ⊆S. Equivalently, the simplexwWJ hascotype J.
We have already seen the canonical type function in Section 1.6.2 in the context of finite reflection groups, where we also saw examples illustrating it.
Here is one more:
Example 3.7.Let W be the group of isometries of the plane generated by the (affine) reflections with respect to the sides of an equilateral triangle. This is an example of a Euclidean reflection group. Although we will not treat the theory of such groups systematically until Chapter 10, the reader should find it plausible thatW is the Coxeter group
s, t, u;s2=t2=u2= (st)3= (tu)3= (su)3= 1
and that the Coxeter complex Σ
W,{s, t, u}
is the plane tiled by equilat- eral triangles. We will give an ad hoc proof of this in Section 3.4.2 below (Example 3.76); in the meantime, the reader is advised to take the assertion on faith. Figure 3.1 shows the panels of the fundamental chamberC labeled by the reflections that fix them, or, equivalently, by their cotypes. The black vertex ofC is not in the panel fixed bys, so it is of typesand hence all black vertices are of types. Similar remarks apply to the other two types.
Exercises
Throughout these exercises (W, S) is a Coxeter system and Σ:=Σ(W, S) is the associated Coxeter complex.
3.8.Give an alternative proof thatΣ is simplicial based on Exercise A.3.
3.9.The canonical type function yields a notion ofs-adjacency for anys∈S (Section A.1.4). Show that this is consistent with Definition 3.3.
C
s t
u
Fig. 3.1.The canonical type function; black =s, gray =t, white =u.
3.10.For every simplexA∈Σ, show that A=
C≥A
C , whereC ranges over the chambers ≥A.
3.11. (a) Let C andD be chambers ofΣ such thatd(C, D)≤d(C, D) for every chamber D adjacent to D. Show thatΣ is spherical and that C andDare opposite. [Recall that ifΣis spherical, thenΣcan be identified with the complex associated to a finite reflection group, so “opposite”
makes sense.]
(b) Deduce (or show directly) thatΣ is spherical if and only if it has finite diameter, where thediameter of a chamber complex is the supremum of the gallery distances between its chambers.
3.12.Assume that (W, S) is irreducible and W is infinite. The content of Proposition 2.43, then, is thatΣ has infinitely many vertices of each types.
[Take J=S{s}in the proposition.]
(a) Deduce that for each chamberC and eachs∈S, the distanced(C, y) is unbounded as y ranges over the vertices of types. Equivalently, d(C, y) is unbounded for a fixed vertexyof typesasCranges over all chambers.
Hered(−,−) denotes the gallery distance defined in Section A.1.3.
(b) Use Proposition 2.45 to prove the following stronger result: For each vertex x ∈ Σ and each s ∈ S, the distance d(x, y) is unbounded as y ranges over the vertices of types.