A question on reflection of Coxeter group

Question

Give a Coxeter system $(W,S)$, it is very easy to check that every elements of the form $wsw^{-1}$ is a reflection. However it seems a little hard when I try to prove that every reflection in $W$ is a conjugate of a simple root. Could anyone give me some help, please?

Answer 1

Your question risks becoming tautological, as usually we define the reflections of $(W,S)$ to be the conjugates of $S$. To avoid that problem I'm going to assume that you have a discrete action of $W$ on some space where the elements of $S$ act by reflections, and the mirrors of these simple reflections bound a fundamental domain (ie the action of $W$ corresponds to the Coxeter system $(W,S)$ rather than to some other Coxeter system $(W,S')$. This could be the action on $\mathbb{R}^{|S|}$ given by the Tits representation, or the action on the Coxeter complex, or on the Davis complex for example. There first two actions are essentially identical, and there's not too much difference between the Coxeter and Davis complexes for this propose, so I'm going to opt to work in the Coxeter complex $X$ of $(W,S)$ if that's OK.

Consider an element $r$ which acts as a linear reflection in some codimension 1 hyperplane $H=\textrm{Fix}(r)$ (here $\textrm{Fix}$ denotes the set of fixed points of an element). Let $C$ be a chamber of $X$ which has $H$ as a wall. If $C_0$ is the fundamental chamber of $X$, there is a unique element $w\in W$ such that $wC_0=C$. Then $w^{-1}H$ is a wall of $C_0$, so by the definition of the Coxeter complex, $w^{-1}H$ is the mirror of some simple reflection $s\in S$. But it is easy to check that $\textrm{Fix}(r)=C=wC_0=w\textrm{Fix}(s)=\textrm{Fix}(wsw^{-1})$. Since the action is faithful, we can conclude that $r=wsw^{-1}$.