Exercise 5, § 1

Let $X$ be a non-empty set and $W$ a set of permutations of $X$. Assume given a set $\mathscr{R}$ of equivalence relations on $X$, an element $x_0 \in X$ and a map $\varphi : H \mapsto s_H$ from $\mathscr{R}$ to $W$. Denote by $\mathscr{R}_0$ the set of $H \in \mathscr{R}$ such that $s_H(x_0) \equiv x_0 \mod. H'$ for all $H' \neq H$ in $\mathscr{R}$, and by $S_0$ the set of $s_H$ for $H$ in $\mathscr{R}_0$. We make the following assumptions:

(i) For any $H \in \mathscr{R}$, there are two equivalence classes modulo $H$ that are permuted by $s_H$ and $s_H^2 = 1$.

(ii) For all $H \in \mathscr{R}$ and all $w \in W$, the transform $w(H)$ of $H$ by $w$ is an equivalence relation belonging to $\mathscr{R}$ and $s_{w(H)} = ws_Hw^{-1}$.

(iii) For any $w \neq 1$ in $W$, the set of $H \in \mathscr{R}$ such that $w(x_0) \not\equiv x_0 \mod. H$ is finite and meets $\mathscr{R}_0$.

a) Prove that $(W, S_0)$ is a Coxeter system (use Prop. 6 of no. 7).

b) Prove that the length $l_{S_0}(w)$ is equal to the number of elements $H \in \mathscr{R}$ such that

c) Let $E$ be a finite set and $X$ the set of total order relations on $X$. Denote by $W$ the group of permutations of $E$, acting in the obvious way on $E$. Let $i$ and $j$ be distinct elements of $E$; say that two elements $R$ and $R'$ of $X$ are equivalent mod. $H_{ij}$ if either $R(i, j)$ and $R'(i, j)$ or $R(j, i)$ and $R'(j, i)$; and denote the transposition of $i$ and $j$ by $s_{ij}$. Let $\mathscr{R}$ be the set of equivalence relations of the form $H_{ij}$ and $\varphi$ the map from $\mathscr{R}$ to $W$ defined by $\varphi(H_{ij}) = s_{ij}$; finally let $x_0$ be an arbitrary element of $X$. Show that the objects thus defined satisfy assumptions (i) to (iii), and recover the results of Exerc. 4.