# When does a finite irreducible Coxeter Group act on the cosets of a parabolic subgroup faithfully?

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?

• Shouldn't this happen for any proper $J\subset S$? If $g\in W$ fixes every coset of $W_J$ then $g$ is in every conjugate of $W_J$, and any intersection of parabolic subgroups is parabolic (i.e., conjugate to a standard parabolic), so the intersection of all conjugates of $W_J$ is a proper normal parabolic subgroup. But $(W,S)$ is finite and irreducible, so the only proper parabolic subgroup that's normal is $\{1\}$, hence $g=1$. Jun 7, 2021 at 16:40

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.)

• If $W$ is allowed to be infinite, probably you have to write "Any finite intersection of parabolic... is parabolic". Then you probably use (in an arbitrary Coxeter group) that if $K$ is a subset of $S$ and is not a union of components, then for $w\in S-K$ adjacent to $K$, $w$ doesn't normalize $W_K$.
– YCor
Jun 18, 2021 at 12:08
• Ah yes, good point, the infinite case is a little more complicated. Jun 18, 2021 at 12:49