Finite Reflection Groups
1.6 Special Properties of Σ
1.6.2 Σ Is a Colorable Chamber Complex
It is immediate that the simplicial complexΣassociated to a finite reflection group is a chamber complex as defined in Section A.1.3. And, using barycentric subdivisions as motivation again, we already know thatΣ is often colorable.
In fact, it is always colorable.
Proposition 1.128.The chamber complexΣ associated with a finite reflec- tion groupW is colorable.
Proof. We will use the criterion in terms of retractions stated at the end of Section A.1.3. Choose a chamber C. Then we can define φ:Σ → Σ≤C by letting φ(A) be the unique face of C that is W-equivalent to A (see Theo- rem 1.104). It is easy to check that φ is a well-defined chamber map and a
retraction.
It is clear from the proof that two simplices have the same type (or color) if and only if they are in the sameW-orbit. In particular:
Corollary 1.129.The action of W on Σis type-preserving.
Recall from Section A.1.3 that the (essentially unique) type function onΣ is completely determined once one assigns types to the vertices of a “funda- mental” chamber C. There is a canonical choice in which the set of colors is the setSof fundamental reflections. Namely, for each vertexvofC, the panel of C not containing v is fixed by a unique reflection s, and we declarev to have types. More succinctly, the panel ofCfixed byshas cotypes, i.e., it is ans-panel. See Figure 1.13.
s C A sC s
Fig. 1.13.Ais thes-panel ofC.
We can also describe the type function by means of the correspondence between simplices and standard cosets (Theorem 1.111): The simplex corre- sponding to a cosetwShas cotypeS.
Figure 1.14 shows the canonical type function when W is the reflection group of type A2(see also Figure 1.9). HereΣ is combinatorially a hexagon.
Our definition implies that the white vertex of the fundamental chamber has type s; hence all of the white vertices have type s. Similarly, all the black vertices have typet.
Hs C
Ht
tC
sC
Fig. 1.14.The canonical type function;◦=s,•=t.
1.6 Special Properties ofΣ 61 The reader might find it instructive to work out the types of the three vertices of the fundamental chamber in Figure 1.12. For example, the vertex with stabilizers, thas typeu.
The fact that the setSplays a dual role, being both a subset ofW and a set of types of vertices ofΣ, is potentially confusing. In practice, however, it turns out to be quite useful. As an illustration of the dual role, consider the opposition involution of Σ =Σ(W, V), which is by definition the simplicial automorphism A→ −A forA ∈Σ. We denote it by opΣ. Thinking ofS as the set of types of vertices, we get an inducedtype-change involution (opΣ)∗ (possibly trivial) by Proposition A.14. On the other hand, thinking of S as a subset of W, we have an involution σ0 of S, given by conjugation by the longest elementw0(Section 1.5.2). These two involutions turn out to coincide:
Proposition 1.130.The type-change map(opΣ)∗ isσ0.
Proof. Fix s ∈ S and let A be the panel of C of cotype s, i.e., the panel of C fixed by s. We have to show that the cotype of the panel −A of −C is σ0(s). In view of Corollary 1.129, the cotype of−A is the same as that of w0(−A) = −w0(A). The latter is the panel ofC fixed by w0sw0 =σ0(s), so
it does indeed have cotypeσ0(s).
Corollary 1.131.If (W, V) is essential, then the following conditions are equivalent:
(i)The involution σ0 is trivial.
(ii)w0 is central inW.
(iii)The opposition involution of Σ=Σ(W, V)is given by the action of w0. (iv)w0=−1.
(v)W contains−1.
(vi)w0D=−D for every chamberD.
If (W, S)is irreducible, these conditions are also equivalent to:
(vi)W has a nontrivial center.
Proof. From the original definition ofσ0, we see that it is trivial if and only if w0 commutes with each s ∈ S. Hence (i) and (ii) are equivalent. On the other hand, Proposition 1.130 shows that (i) holds if and only if opΣ is type- preserving. But this holds if and only ifw0 acts onΣas opΣ. [If−1 is type- preserving, then it agrees withw0on the vertices of the fundamental chamber;
moving out along galleries, one deduces that it agrees with w0 on the entire chamber complexΣ.] Thus (i) and (iii) are equivalent.
Next, the fact that (W, V) is essential implies that ifw0and opΣ agree as simplicial maps, thenw0=−1 as a linear map onV. This follows, for example, from the fact that the vertices ofΣcan be identified with a set of unit vectors that spanV (see Section 1.5.8). Hence (iii) and (iv) are equivalent. And (iv) is equivalent to (v) becausew0is the unique element ofW that takesCto−C.
Turning now to (vi), we can argue that it is equivalent to (iii) because two simplicial automorphisms that agree on all chambers must agree on all panels and hence on all vertices. Alternatively, (vi) is equivalent to (ii) because (vi) says thatw0 is independent of the choice of fundamental chamber and hence is invariant under conjugation.
Finally, if (W, S) is irreducible, then we have already shown in Corol- lary 1.91 (see also Exercise 1.102) that the center of W is nontrivial if and
only if (v) holds.
Exercises
1.132.Give an example to show that we cannot drop the assumption that (W, V) is essential in the corollary.
1.133.Using the classification of finite reflection groups (Section 1.3), find the involution σ0 for as many of them as you can. [See Section 5.7 for the complete list.]