Finite Reflection Groups
1.5 The Simplicial Complex of a Reflection Group
1.5.8 The Poset Σ as a Simplicial Complex
The fact that every chamber is a simplicial cone in the essential case suggests that the hyperplanes inHcut the unit sphere inV into (spherical) simplices.
Thus it seems intuitively clear that the poset Σ := Σ(W, V) of cells can be identified with the poset of simplices of a simplicial complex that triangulates a sphere of dimension rank(W, V)−1. In this subsection we prove this state- ment rigorously. Before proceeding, the reader might find it helpful to look at the first few paragraphs of Appendix A, where we explain our conventions regarding simplicial complexes. In particular, the statement of the following proposition has to be understood in terms of Definition A.1.
Proposition 1.107.The posetΣ is a simplicial complex.
Proof. We may assume that (W, V) is essential, sinceΣ remains unchanged, up to canonical isomorphism, if we pass to the essential part. According to Definition A.1, we must check two conditions. Condition (a) is that any two elements of Σ have a greatest lower bound; this has already been proved in Proposition 1.47. As to Condition (b), concerning the poset Σ≤A of faces of a cellA∈Σ, we know thatAis a face of a chamber, so it suffices to consider the case thatAis a chamber. But it is a trivial matter to compute the poset of faces of a simplicial cone, and this poset is indeed isomorphic to the set of
subsets of{1, . . . , n}.
We gave this somewhat abstract proof of the proposition in order to intro- duce the unorthodox terminology that we use regarding simplicial complexes;
this will be useful later. But it is easy to chase through the discussion in Sec- tion A.1.1 in order to describe in more conventional terms how, in the essential case, Σ can be identified with an abstract simplicial complex (in which the simplices are certain finite subsets of a set of “vertices”):
Every 1-dimensional cell A ∈ Σ is a ray R∗+v, where R∗+ is the set of positive reals andv is a unit vector; the unit vectorsv that arise in this way are the vertices of our simplicial complex. For each (q+ 1)-dimensional cell A ∈ Σ (q ≥ −1), there is a q-simplex {v0, . . . , vq} in our complex, where thevi are the unit vectors in the 1-dimensional faces ofA. It should be clear that we do indeed obtain a simplicial complex in this way and that Σ can
1.5 The Simplicial Complex of a Reflection Group 53 be identified with the poset of simplices of this complex. Notice that we have allowed q =−1 above. The cellA is {0} in this case, and it corresponds to the empty set of vertices. [Our convention, as explained in Section A.1.1, is that the empty set is always included as a simplex of an abstract simplicial complex.]
Proposition 1.108.The geometric realization|Σ|is canonically homeomor- phic to a sphere of dimension rank(W, V)−1.
Proof. Again we may assume that (W, V) is essential, in which case we will exhibit a homeomorphism from|Σ|to the unit sphere inV. Recall from Sec- tion A.1.1 that |Σ| consists of certain convex combinations
vλvv, wherev ranges over the vertices of Σ, viewed as basis vectors of an abstract vector space. Now the vertices v0, . . . , vq of any A∈ Σ can also be viewed as unit vectors inV, and as such, they are linearly independent. Hence we have a map
|Σ| →V {0}, given by
λvv →
λvv. Composing this with radial pro- jection, we obtain a continuous mapφ:|Σ| →Sn−1. Since φtakes|A| ⊂ |Σ| bijectively to A∩Sn−1 ⊂V, it is bijective and therefore a homeomorphism
(by compactness of|Σ|).
In view of the results of this section, an essential finite reflection group of ranknis also called aspherical reflection groupofdimension n−1.
Exercise 1.109.SupposeW is the group of symmetries of a regular solidX.
Make an intelligent guess as to how to describeΣ directly in terms ofX. 1.5.9 A Group-Theoretic Description of Σ
We started the chapter with a “concrete” groupW, given to us as a group of linear transformations (or, in more geometric language, as a group of isome- tries of Euclidean space, or, even better, as a group of isometries of a sphere).
The geometry gave us, after we chose a fundamental chamberC, a setSof gen- erators ofW. The geometry also gave us a simplicial complexΣ:=Σ(W, V), constructed by means of hyperplanes and half-spaces. We will prove below, however, that if we forget the geometry and just viewW as an abstract group with a given set S of generators, then we can reconstruct Σ by pure group theory. This observation will have far-reaching consequences. For simplicity, we assume in this subsection that (W, V) isessential.
Consider first the subcomplex Σ≤C consisting of the faces of the funda- mental chamber C. To every face A ≤C, we associate its stabilizer WA. In view of Theorem 1.104 and Corollary 1.105, WA is also the stabilizer of any point x ∈ A, and it fixes A pointwise. The theorem also says that WA is generated by a subset of our given generating setS. Subgroups of this form have a name:
Definition 1.110.A subgroup ofW is called astandard parabolic subgroup, or simply astandard subgroup, if it is generated by a subset ofS. Any conjugate of such a subgroup will be calledparabolic, without the adjective “standard.”
Thus we have a function φ from Σ≤C to the set of standard subgroups of W, and we will show that φ is a bijection. In fact, we can construct the inverseψofφby taking fixed-point sets: LetWbe a standard subgroup ofW, generated by a setS ⊆S; then the fixed-point set of W inC is obtained by intersecting C with the walls ofC corresponding to the reflections in S. So this fixed-point set is equal toAfor someA≤C, and we can setψ(W) :=A.
Using the stabilizer calculation in Section 1.5.7, one can easily check that ψ is inverse toφ.
Note next that φ and its inverse ψ are order-reversing. For ψ, this is immediate from the definition. In the case of φ, the assertion follows from the fact thatWA fixesA pointwise and hence stabilizes every face ofA. We therefore have a poset isomorphism
Σ≤C∼= (standard subgroups)op, (1.24) where “op” indicates that we are using the opposite of the usual inclusion order. We will also describe the poset on the right in (1.24) as the poset of standard subgroups, ordered by reverse inclusion. Figure 1.12 illustrates this isomorphism whenn= 3 andS={s, t, u}. HereCis the cone over a triangle, and we have drawn a sliceT ofC(or, equivalently, the intersection ofCwith the unit sphere). The figure shows the stabilizer of almost every face of T, the one exception being the empty face, which is not visible in the picture.
The empty face corresponds to the cellA:={0}, which would appear in the picture if we drew the whole chamber C instead of justT. It is the smallest face of T, and its stabilizer is the largest standard subgroup of W, namely, W itself. Similarly, the largest face isT itself, whose stabilizer is the smallest standard subgroup {1} (generated by∅ ⊂S).
s, t
{1} s t
s, u u t, u
Fig. 1.12.The standard parabolic subgroups as stabilizers.
Returning now to the general case, we can use the W-action to extend our isomorphism to one from the whole posetΣ to the set ofstandard cosets in W, i.e., the cosets wW of standard subgroups. Indeed, we can send a typical elementwA ∈Σ (w∈W, A≤C) to the cosetwWA. It is a routine matter to deduce the following result from what we did above forΣ≤C: Theorem 1.111.There is a poset isomorphism
1.5 The Simplicial Complex of a Reflection Group 55 Σ∼= (standard cosets)op
that is compatible with theW-action, whereW acts on the cosets by left trans-
lation.
We can express the theorem more briefly by saying thatΣisW-isomorphic to the poset of standard cosets inW, ordered by reverse inclusion.
Exercise 1.112.Let W be the reflection group of type An−1 (symmetric group on nletters), with the standard choice of fundamental chamber. Thus S is the set{s1, . . . , sn−1}of basic transpositions, wheresiinterchangesiand i+ 1. We stated without proof in the discussion of Example 1.10 that the complexΣ associated toW is the barycentric subdivision of the boundary of an (n−1)-simplex. (See also Exercise 1.109.) Prove this rigorously.