# About parabolic subgroup of a Weyl group

Let $$W$$ be a Weyl group/Coxeter group. Let $$\Phi$$ be the associated root system, fix a positive root system $$\Phi^+$$ and let $$\Delta$$ be the set of simple roots.

Let $$W_I$$ be the parabolic subgroup of $$W$$ generated by $$I\subseteq \Delta$$.

1. Does $$W_I=W_J\implies I=J$$?

2. Does $$s\in W$$ with $$s\neq 1$$, $$s^2=1\implies s=s_\alpha$$ for some root $$\alpha$$?

• What are your thoughts? What properties of Weyl groups do you know which might help? Also, in question 2 I assume you want to exclude the trivial counterexample $s=id$. Commented Dec 29, 2018 at 4:51
• My thought is that if 2. is correct. Then I can say the set of order 2 elements in $W_I$ is the set of order 2 elements in $W_J$. Then $\{s_\alpha:\alpha\in\Phi_I\}=\{s_\alpha:\alpha\in\Phi_J\}$. This implies $\Phi_I=\Phi_J$ and then $I=J$. Commented Dec 29, 2018 at 17:55
• $s=-id$ is an element of the Weyl group for many root systems (e.g. all of type $B_n, C_n$, and all exceptional ones except $E_6$), and is a counterexample to 2. as soon as the rank is $\ge 2$. Commented Jan 2, 2019 at 1:57
• 1. is true: the parabolic subgroups are in bijection with subsets of $\Delta$. Commented Feb 22, 2022 at 17:25