Suppose you have a Coxeter group $W$ of type $I_2(m)$, the dihedral group.
If $m$ is odd, and $s_1$ and $s_2$ are the simple generators, $w_0$ the longest element, is it true that $w_0s_1w_0=s_2$ and, equivalently, $w_0s_2w_0=s_1$. And if $m$ is even, is it true that $w_0s_1w_0=s_1$, and $w_0s_2w_0=s_2$?
I know that when $W$ acts as reflections on a Euclidean space, $w_0$ is central when $m$ even, and acts as $-1$, and when $m$ is odd, $w_0$ is not central acts as the negative of the nontrivial automorphism of the Coxeter diagram, so I am wondering if there are similar things we can say when doing multiplication in the group $W$ itself, instead of viewing it as acting on some Euclidean space.