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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 …

Let W be a finite Coxeter group and $\Phi^+$ the set of its positive roots. The Coxeter group acts on $\Phi^+$ by $(w, \alpha) \mapsto w \cdot \alpha$ if $w \cdot \alpha \in \Phi^+$ and $(w, \alpha) \ …

The Chevalley basis of $sp_4$ is generated by \begin{align} & e_1=\left(\begin{array}{cccc} 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & -1 & 0 \end{array}\right), \ e_2 = \left(\begin{arr …

The longest word in type $A_3$ Weyl group written as a matrix is \begin{align} w_0=s_1s_2s_1=\left(\begin{array}{cccc} 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0\\ 0 & 1 & 0 & 0\\ 1 & 0 & 0 & 0 \end{array}\right) …

Let $W$ be a Weyl group generated by the simple reflections $s_i$, $i \in I$, where $I$ is the vertex set of the Dynkin diagram of $W$. For $J \subset I$, let $W_J$ be the subgroup of $W$ generated by …