For any finite crystallographic reflection group $W = \langle s_1, \ldots , s_n\rangle$, every hyperplane reflection is of the form $ws_iw^{-1}$ for some $i$ and some $w \in W$.

A finite crystallographic reflection group $W$ is a Coxeter group with the presentation \begin{align}\label{Coxeter system} S=\langle s_1, s_2, \ldots, s_n \mid (s_is_j)^{m_{ij}} = 1 \rangle, \end{align} where $(s_i)_{1 \leq i \leq n}$ is the set of simple reflections and $m_{ij} \in \{2,3,4,6\}$. The pair $(W,S)$ is called a Coxeter system.

I have some questions:

- Is it true that every finite reflection group consists of some (not necessarily hyperplane) reflections and some rotations?
- What are the reduced words of reflections under a Coxeter system $(W,S)$?

For any finite reflection group, the number of hyperplane reflections is the number of positive roots in the corresponding root system, see section 1.14 of Jim Humphreys' book "Reflection groups and Coxeter groups".

somereduced word of that form (palindromic). This paper studies reduced words for reflections in Coxeter groups in detail: deepblue.lib.umich.edu/handle/2027.42/46149 $\endgroup$ – Sam Hopkins Dec 20 '16 at 20:355more comments