There is a symmetric group action on the power set of positive roots in type A. The action is defined as follows.
Denote by $\alpha_1, \ldots, \alpha_n$ be the set of simple roots in a root system. In type $A$, positive roots are of the form $\alpha_i + \cdots + \alpha_j$ ($1 \le i \le j \le n$). Identify a positive root $\alpha_i + \cdots + \alpha_j$ with a pair $(i, j)$. Let $\mu=\{ \mu_1, \ldots, \mu_m \}$ be set of positive roots.
Define ${\bf i}_\mu=\text{sort}(\min(\mu_1), \ldots, \min(\mu_m))$, ${\bf j}_\mu=\text{sort}(\max(\mu_1), \ldots, \max(\mu_m) )$ (ordered from small to large).
Denote ${\bf p}_\mu = \{ (\min(\mu_i), \max(\mu_i)): i \in [m]\}$ (as a multi-set).
For ${\bf c}=(c_1, \ldots, c_m), {\bf d} = (d_1, \ldots, d_m) \in \mathbb{Z}^m$, we denote $${\bf p}_{{\bf c}, {\bf d}} = \{ (c_i, d_i): i \in [m] \} \quad \quad (1)$$ (as a multi-set), where we ignore $(c_i, d_i)$ if $c_i>d_i$.
Let $S_m$ be the symmetric group on $\{1, \ldots, m\}$. For ${\bf i} = (i_1, \ldots, i_m) \in \mathbb{Z}^m$, denote by $S_{\bf i}$ the subgroup of $S_m$ consisting of elements $\sigma$ such that $i_{\sigma(j)}=i_j$, $j \in [m]$.
The following result is obvious.
Lemma For ${\bf i}, {\bf j} \in \mathbb{Z}^m$, $w,w' \in S_m$, we have that ${\bf p}_{w' \cdot {\bf i}, {\bf j}} = {\bf p}_{w \cdot {\bf i}, {\bf j}}$ if and only if $w' \in S_{{\bf j}}wS_{{\bf i}}$.
Is there similar result for other Weyl groups in the literature?
In type $A$, the map ${\bf p}_{{\bf i}, {\bf j}} \mapsto {\bf p}_{w\cdot {\bf i}, {\bf j}}$ sends a subset of positive roots to a subset of positive roots. In other types, is there a similar action of a Weyl group on the power set of positive roots?
Thank you very much.
