The <Cayley graph> $\operatorname{Cay}_S(W)$ has vertex set $W$ and an unoriented edge $\{w,ws\}$ for every $w\in W$ and $s\in S$.

For a <reflection of a Coxeter group> $r\in R$, its wall is
$$
H_r=\bigl\{\{w,ws\}:wsw^{-1}=r\bigr\}.
$$
Equivalently, these are the edges fixed setwise and reversed by left multiplication by $r$. Removing their interiors separates the Cayley graph into two <half-space of a Coxeter group>[half-spaces].

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 111 / 2 / a / Solution

A reflection of a Coxeter system $(W,S)$ is a conjugate $wsw^{-1}$ of a simple generator $s\in S$.


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 Incoming links: Reflection of a Coxeter group