Exercise 16, § 1

Let $(W, S)$ be a Coxeter system. For $s \in S$, denote by $W^{(s)}$ the subgroup $W_{S-\{s\}}$ of W, by A the set of subsets of W of the form $w W^{(s)}$ for $w \in W$ and $s \in S$, and by $\mathscr{C}$ the set of subsets of A of the form $C_w = \{ w W^{(s)} \mid s \in S \}$ for $w \in W$, which we call the chambers of A (Exerc. 15).

a) Show that the map $w \mapsto C_w$ is a bijection from W onto $\mathscr{C}$.

b) Show that two distinct chambers $C_w$ and $C_{w'}$ are adjoining if and only if there exists $s \in S$ such that $w' = ws$. Deduce that A (together with $\mathscr{C}$) is an apartment (Exerc. 15), which we call the apartment of $(W, S)$. Show that the length of a minimal gallery with ends $C_w$ and $C_{w'}$ is equal to $l_S(w^{-1} w')$.

c) Let $\mathscr{F}$ be the set of facets of A and let $F \in \mathscr{F}$. Show that there exist a unique subset X of S and an element $w \in W$ such that $w W_X = \bigcap_{i \in F} i$. Then F is said to be of type X. Show that the codimension of F is equal to the cardinal of X. Show that the map $j : F \mapsto \bigcap_{i \in F} i$ is a strictly decreasing bijection (with respect to inclusion) from $\mathscr{F}$ to the set of subsets of W of the form $w W_X$ for $w \in W$ and $X \subset S$. Show that every facet of type X contains a unique facet of type Y for every Y such that $X \subset Y \subset S$.

d) Let W act on A be left translations and put C = C_e. Show that W acts simply-transitively8 on $\mathscr{C}$. Let $C_1, \ldots, C_n$ be chambers of A. Show that the following conditions are equivalent:

(i) the sequence $\Gamma = (C_0 = C, C_1, \ldots, C_n)$ is an injective gallery;

(ii) there exists a sequence $s = (s_1, \ldots, s_n)$ of elements of S such that $C_j = t_j(C_{j-1})$ for $1 \leq j \leq n$, where $t_j$ is the element of $\Phi(s)$ defined in no. 4, formula (11).

Show that if these conditions are satisfied, the sequence-s is unique and $C_n = s_1 \ldots s_n(C)$. Show that the gallery $\Gamma$ is minimal if and only if the sequence $s(\Gamma) = s$ is a reduced decomposition of $w = s_1 \ldots s_n$.

e) Let T be the union of the conjugates of S. For $t \in T$, the set $L_t$ of points of A invariant under t is called the wall defined by t. Show that $L_t$ is a union of panels and that a panel F is contained in $L_t$ if and only if $j(F)$ is of the form $wW_{\{s\}}$ with $t = wsw^{-1}$. Deduce that, for any panel F, there is a unique element $t = t(F) \in T$ such that $F \subset L_t$: $L_t$ is called the support of F.

Show that if $w(L_t) = L_t$ (for $w \in W$), then $w = 1$ or $w = t$.

f) Let $w', w'' \in W$. Put $C' = w'(C), C'' = w''(C)$ and let $\Gamma = (C_0 = C', C_1, \ldots, C_n = C'')$ be an injective gallery with ends $C'$ and $C''$. Let $t_j$ be the element of T defining the support wall of the panel $C_j \cap C_{j-1}$ (for $1 \leq j \leq n$). Show that the sequence $\Psi(T) = ({w'}^{-1}t_jw')_{1 \leq j \leq n}$ coincides with the sequence $\Phi(s({w'}^{-1}(\Gamma)))$. For $t \in T$, let $n(\Gamma, t)$ be the number of times ${w'}^{-1}tw'$ appears in $\Psi(\Gamma)$. Deduce from Lemma 1 of no. 4 that the number $(-1)^{n(\Gamma, t)}$ depends only on $C'$ and $C''$ and not on $\Gamma$: we denote it by $\eta(C', C'', t)$. Show that the relation $\eta(C', C'', t) = 1$ is an equivalence relation between $C'$ and $C''$ that has two equivalence classes that are permuted by t. Denote by $\mathscr{C}^+(t)$ the equivalence class containing C and by $\mathscr{C}^-(t)$ the other.

Show that, for $w \in W$ and $s \in S$, the chamber $w(C)$ belongs to $\mathscr{C}^+(s)$ if and only if $l(sw) > l(w)$.

g) Let $A^+(t)$ (resp. $A^-(t)$) be the union of the chambers belonging to $\mathscr{C}^+(t)$ (resp. $\mathscr{C}^-(t)$) (for $t \in T$). Show that $A^+(t) \cap A^-(t) = L_t$. (To show that $A^+(t) \cap A^-(t) \subset L_t$, reduce to the case $t \in S$. If $a \in A^+(t) \cap A^-(t)$, put $a = wW^{(s)}$ with $s \in S$ and $w$ being ($\varnothing, S - \{s\}$)-reduced (Exerc. 3). If $w(C) \in \Sigma^-(t)$, then $l(tw) < l(w)$ and $w = ts_1 \ldots s_q$ with $s_j \in S$ and $l(w) = q + 1$. Since $a \in A^-(t)$, there exists $x \in W^{(s)}$ such that $wx(C) \in \mathscr{C}^+(t)$. Then

but since $twx = s_1 \ldots s_q x$ this is a contradiction. Thus, $w(C) \in \mathscr{C}^+(t)$ and $l(tw) = 1 + l(w)$. If $x \in W^{(s)}$ is such that $l(twx) < l(wx)$, deduce from Exerc. 1 that $twx = wx'$ with $x' \in W^{(s)}$, and hence that $ta = a$ and $a \in L_t$.

8 Let H be a group and E a non-empty set on which H acts. H is said to act simply-transitively on E if the map $h \mapsto h.x$ is a bijection from H onto E for all $x \in E$; then E is said to be a principal homogeneous set for H.

The subsets $A^+(t)$ and $A^-(t)$ are called the halves of $A$ determined by the wall $L_t$. Two points of $A$ are said to be on the same side (resp. strictly on opposite sides) of $L_t$ if they belong (resp. do not both belong) to one of the two halves. Any facet is contained in one of the halves determined by $L_t$. If two facets are contained in different halves, they are said to be on opposite sides of $L_t$, or to be separated by $L_t$.

h) Let $w \in W$. Show that $l(w)$ is equal to the number of walls separating $C$ and $w(C)$.

i) Show that the map $\varphi$ that takes the half $A^+(t)$ (resp. $A^-(t)$) to $(1, t)$ (resp. $(-1, t)$) is a bijection from the set $\mathscr{M}$ of halves of $A$ onto the set $R = \{1, -1\} \times T$ (cf. no. 4). With the notation of Lemma 1 of no. 4, show that $\varphi(w(M)) = U_w(\varphi(M))$ for all $w \in W$ and $M \in \mathscr{M}$.