Let $(W,S)$ be a finite and irreducible Coxeter Group. For $J \subseteq S$, let $W\_J = \langle s | s \in J \rangle$, a parabolic subgroup. For which $J$ is the action (group multiplication on the left) of $W$ permuting the (left) cosets of $W\_J$ faithful exactly?


Is there a good reference to find this result in the literature?



When does a finite irreducible Coxeter Group act on the cosets of a parabolic subgroup faithfully?
Stumbled on this again, I should make my comment an official answer.


This is true for any proper $J\subseteq S$. Suppose $g\in W$ fixes every coset of $W\_J$, so $g$ lies in the intersection of all conjugates of $W\_J$. Any intersection of parabolic subgroups (meaning conjugates of standard parabolic subgroups) is a parabolic subgroup, so the intersection of all conjugates of $W\_J$ is a proper normal parabolic subgroup of $W$. But $(W,S)$ is finite and irreducible, so the only proper normal parabolic subgroup of $W$ is $\{1\}$. Hence $g=1$, i.e., the action is faithful.


(As a remark, I don't think this used finiteness of $W$ (??), just irreducibility of $(W,S)$. The key is just that there are no proper non-trivial normal parabolic subgroups.)


