Exercise 4, § 1

Let $n$ be an integer $\geq 2$. For any integer $i$ with $1 \leq i \leq n - 1$, denote by $s_i$ the transposition of $i$ and $i + 1$ in the set $\{1, 2, \ldots, n\}$, and by $H_i$ the set of $w \in S_n$ such that $w^{-1}(i) < w^{-1}(i + 1)$; put $S = \{s_1, \ldots, s_{n-1}\}$. Show that $(S_n, S)$ is a Coxeter system and that $H_i$ is the set of $w \in S_n$ such that $l(w) < l(s_i w)$ (use Prop. 6 of no. 7).