Exercise 2, § 6

Assume that $W$ is irreducible. Let $C$ be a chamber relative to $W$, $(H_1, \ldots, H_l)$ its walls, and $e_i$ a non-zero vector orthogonal to $H_i$. Assume that $e_1, \ldots, e_r$ (resp. $e_{r+1}, \ldots, e_l$) are pairwise orthogonal (cf. no. 2). For all $u \in \mathbf{Z}$, define $H_u$ and $s_u$ by $H_u = H_k$ if $u \equiv k \pmod{l}$ and $s_u = s_{H_u}$.

a) Show that the elements of $\mathfrak{H}$ are the $s_1 s_2 \ldots s_{u-1} H_u$ for $u = 1, 2, \ldots, lh/2$.

b) Let $s' = s_1 \ldots s_r$ and $s'' = s_{r+1} \ldots s_l$, so that $c = s's''$ is the Coxeter transformation associated to the ordered chamber C. Let $w_0$ be the element of W that transforms C to $-C$ (cf. § 4, Exerc. 2). Show that, if $h$ is odd,

c) Put $S = \{ s_1, \ldots, s_l \}$. The pair $(W, S)$ is a Coxeter system. If $w \in W$, denote by $l_S(w)$ the length of $w$ with respect to S (Chap. IV, § 1, no. 1). Show that

Deduce that, with the assumptions in b),

Show on the other hand that $l_S(w_0) = \operatorname{Card}(\mathfrak{H}) = hl/2$ (use Exerc. 22 of Chap. IV, § 1). Deduce that $r = l/2$ and that $(s_1, s_2, \ldots, s_{lh/2})$ is a reduced decomposition of $w_0$.

d) Show that, if $h$ is even, $(s_1, \ldots, s_{lh/2})$ is a reduced decomposition of $w_0 = c^{h/2}$. (Same method.)