Exercise 16, § 4

Let (W, S) be an irreducible, finite Coxeter system, and let $n = \mathrm{Card}(S)$.

a) If (W, S) is not of type $F_4$, show that there exists a subset X of S with $n-1$ elements such that $(W_X, X)$ is of type $A_{n-1}$.

b) Identify W with a subgroup of $\mathbf{GL}(\mathbf{R}^S)$ by means of the canonical representation (Chap. V, § 4). Show that there exists a basis $(e_1, \ldots, e_n)$ of $\mathbf{R}^S$ such that, for every permutation $\sigma \in S_n$, the automorphism of $\mathbf{R}^S$ that transforms $e_i$ to $e_{\sigma(i)}$ for $1 \leq i \leq n$ belongs to W ("Burnside's Theorem"). (When (W, S) is not of type $F_4$, use a); when it is of type $F_4$, remark that W contains a subgroup of type $D_4$ (cf. no. 9), which reduces the problem to the preceding case.)

c) Let E be a subgroup of the group of automorphisms of (W, S). Show that the semi-direct product E.W of E by W embeds in a canonical way in $\mathbf{GL}(\mathbf{R}^S)$. Show that the subgroup of $\mathbf{GL}(\mathbf{R}^S)$ thus defined is generated by reflections, except in the following four cases:

(i) (W, S) is of type $A_n$, $n \geq 4$: the group E is of order 2.

(ii) (W, S) is of type $D_4$: the group E is of order 3.

(iii) (W, S) is of type $F_4$: the group E is of order 2.

(iv) (W, S) is of type $E_6$: the group E is of order 2.

Show that, in cases (i) and (iv), $E.W = \{ \pm 1 \} \times W$. Show that, in case (iii), the group E.W does not leave stable any lattice in $\mathbf{R}^S$.

d) Let $W_1$ be a finite subgroup of $\mathbf{GL}_n(\mathbf{R})$ generated by reflections, and assume that $W_1$ is irreducible and essential. Let G be a finite subgroup of $\mathbf{GL}_n(\mathbf{R})$ containing $W_1$. Show that either G is generated by reflections or is of the form E.W, where E and W are one of the types (i), (ii), (iii), (iv) of c).

(Let W be the group generated by the reflections belonging to G. The group G permutes the chambers relative to W. Deduce, as in § 2, no. 3, that G is of the form E.W as above.)