Exercise 21, § 1

Let (W, S) be a Coxeter system and A the associated apartment (cf. Exerc. 16, of which we retain the notation).

a) Show that there exists a unique numbering of A (called the canonical numbering) for which the type of a facet F is that defined in Exerc. 16 c). We shall always consider A to be equipped with this numbering.

b) Show that the allowed automorphisms of A are the operations of W.

c) Let D be a subset of A containing at least one chamber. Show that the following conditions are equivalent: (i) D is the intersection of the halves of A (Exerc. 16 g)) that contain D; (ii) D is convex; (iii) whenever two facets $F_1$ and $F_2$ are contained in D, the convex hull of $F_1 \cup F_2$ is contained in D; (iv) whenever a chamber $C_1$ and a facet F are contained in D and $(C_1, \ldots, C_n)$ is a gallery of minimal length such that $F \subset C_n$, we have $C_i \subset D$ for $1 \leq i \leq n$.

(To show that (iii) $\Longrightarrow$ (iv), use Exerc. 15 b). To show that (iv) $\Longrightarrow$ (i), argue by contradiction. Let $D'$ be the intersection of the halves of A containing D; let $a \in D' - D$, let $C_0$ be a chamber contained in D and let $(C_0, C_1, \ldots, C_n)$ be a gallery of minimal length such that $a \in C_n$. Then $C_j \subset D'$ for all j. Show that there exists an integer $j$ with $0 \leq j < n$ such that $C_j \subset D$ and $C_{j+1} \notin D$. Let M (resp. $M'$) be the half of A determined by the support wall of the panel $C_j \cap C_{j+1}$ and containing $C_j$ (resp. $C_{j+1}$). Show that $D \notin M$. Let $b \in D \cap (A - M)$ and let $\Gamma = (C_j, C'_1, \ldots, C'_p)$ be a gallery of smallest possible length such that $b \in C'_p$. Then $C'_k \subset D$ for $1 \leq k \leq p$ and $C'_p \subset M'$. Let $\pi$ be the folding of A with image $M'$ (Exerc. 18 c)): then $\pi(C_j) = C_{j+1}$ and the gallery $\pi(\Gamma)$ is not injective (Exerc. 18 a)). If $\Gamma' = (C_{j+1}, C'_2, \ldots, C'_{p-2}, C'_p)$ is the gallery obtained from $\pi(\Gamma)$ by suppressing one of the two equal consecutive chambers, the gallery $(C_j, C_{j+1}, C'_2, \ldots, C'_{p-2}, C'_p)$ is minimal by the definition of $\Gamma$. Deduce from (iv) that $C_{j+1} \subset D$. Contradiction.)