Exercise 3, § 1

Let $(W, S)$ be a Coxeter system and $X, Y$ two subsets of $S$. Let $a \in W$. Show that there exists a unique element $w \in W_X a W_Y$ of minimal length and that every element

can be written in the form $w' = x w y$, with $x \in W_X, y \in W_Y$ and $l(w') = l(x) + l(w) + l(y)$ (take an element of minimal length in $W_X a W_Y$ and use Exerc. 1). An element $w \in W$ is said to be $(X, Y)$-reduced if it is the element of minimal length in the double coset $W_X a W_Y$. Show that if $w$ is $(X, \varnothing)$-reduced then $l(x w) = l(x) + l(w)$ for all $x \in W_X$, and that every element of $W$ can be written uniquely in the form $x w$ where $x \in W_X$ and $w$ is $(X, \varnothing)$-reduced. Show that an element $w \in W$ is $(X, \varnothing)$-reduced if and only if $l(xw) > l(w)$ for all $x \in X$ (write $w$ in the form $yw'$, where $y \in W_X$ and $w'$ is $(X, \varnothing)$-reduced).

Show that $w \in W$ is $(X, Y)$-reduced if and only if $w$ is both $(X, \varnothing)$-reduced and $(\varnothing, Y)$-reduced.