Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 111 / 1 / a / Solution

The deletion condition for involutory generators says that whenever a wordis not reduced, there are indices $i<j$ such that deleting $s_i$ and $s_j$ leaves a word representing the same group element.

The exchange condition for a Coxeter group says that if $w=s_1\cdots s_m$ is reduced and $s\in S$ satisfies ${\ell }_S(sw)<{\ell }_S(w)$, thenfor some $i$. There is an equivalent right-handed form for ${\ell }_S(ws)<{\ell }_S(w)$.