Exercise 11, § 2

Let $(W, S)$ be a Coxeter system and I a building numbered by S ($§ 1$, Exerc. 20). If $\Gamma = (C_0, \ldots, C_n)$ is an injective gallery, the type of $\Gamma$, denoted by $s(\Gamma)$, is the sequence $(s_1, \ldots, s_n)$ of elements of S such that the panel $C_{i-1} \cap C_i$ is of type $S - \{s_i\}$ (for $1 \leq i \leq n$). If conditions (WI 1) and (WI 2) of Exerc. 10 are satisfied, I is called a $(W, S)$-*building*.

Let I be a spacious $(W, S)$-building (cf. Exerc. 10 g)) and let G be a group of allowed automorphisms of I satisfying the following condition:

(G₀) Given three distinct chambers C, C' and C'' containing the same panel, there exists an element g ∈ G such that g(C) = C and g(C') = C''.

Choose a chamber C of I and denote by B the stabilizer of C in G.

a) Show that G acts transitively on the set of chambers of I.

b) Let C' and C'' be two chambers. Show that t(C, C') = t(C, C'') if and only if there exists b ∈ B such that C'' = b(C') (if t(C, C') = t(C, C'') = w, consider a reduced decomposition s of w and a minimal gallery Γ' (resp. Γ'') with extremities C and C' (resp. C'') and of type s. Argue by induction on l(w) using (G₀) and a)). Deduce that there exists a bijection w ↦ B(w) from W to $B\backslash G/B$.

c) Let F be the facet of C of type S - {s}. Show that the stabilizer of F in G is equal to B ∪ B(s) (if g(F) = F, then t(C, g(C)) = 1 or s). Show that B ⊂ B(s)B(s) (use (G₀)).

d) Show that B is a Tits subgroup of G and that the Coxeter system of (G, B) is canonically isomorphic to (W, S). (Let w ∈ W and s ∈ S be such that l_S(sw) = l_S(w) + 1. Let g ∈ B(w) and u ∈ B(s); let s = (s_1, ..., s_n) be a reduced decomposition of w, (C = C_0, C_1, ..., C_n = g(C)) a minimal gallery of type s and choose an element s_i ∈ B(s_i); show, by using (G₀), that there exist elements b_i ∈ B such that C_i = b_1 s_1 ... b_i s_i(C). Put C'_i = u(C_{i-1}) and show that the gallery (C, u(C), u(C_1), ..., u(C_n)) is of type (s, s), and is therefore minimal. Deduce that ug ∈ B(sw) and that B(s)B(w) = B(sw). If now l(sw) = l(w) - 1, put w' = sw: then B(s)B(w') = B(w) and hence

finally, since B ⊂ B(s)B(s), we also have B(w') = B(sw) ⊂ B(s)B(w).)

e) Show that there exists a unique isomorphism from the building associated to (G, B) (Exerc. 10) to I, compatible with the action of G and taking the canonical chamber C_e to C.