Notation 2.4.7.3 (Lengths). Let $(W,S)$ be a Coxeter system. Then the group $W$ is generated by $S$: that is, every element of $W$ can be written as a product of elements of $S$. For each $w \in W$, we let $\ell (w)$ denote the smallest nonnegative integer $n$ for which $w$ factors as a product $s_1 s_2 \cdots s_ n$, where each $s_ i$ belongs to $S$. We will refer to $\ell (w)$ as the length of $w$.
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