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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 …

Are there some references about the proof of the following fact? Type $C_n$ Weyl group lies in the centralizer of the longest word $w_0$ in $S_{2n}$. Thank you very much.

Let $V$ be a vector space over $\mathbb{R}$. An element $s \in GL(V)$ is a reflection if $H_s:=\ker(s-1)$ is a hyperplane and $s^2=1$. The eigenvalues of a reflection $s$ are $1, -1$. Every reflection …

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) \ …

Let $W$ be a finite or infinite Coxeter group and $\Phi^+$ the set of its positive roots. In the paper, a subset $A$ of $\Phi^+$ is closed if for all $a, b \in A$, $r_1 a + r_2 b \in \Phi^+$ for som …

Are there some generalized Catalan number of type $B$ such that the sequence of the numbers is $3,9,29,97,333$ for $n=2,3,4,5,6$? As discussed in this previous question, there are at least two types …

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 …

Let $w \in S_n$ and $inv(w) = \{(i,j): i,j \in \{1,\ldots,n\}, i<j, w(i)>w(j)\}$ the inversion set of $w$. Let ${\bf i}=(i_1,\ldots,i_m)$ be a sequence such that $s_{i_1}\cdots s_{i_m}$ is a reduced e …

Dihedral groups are Coxeter groups of type $I_m$, $m \geq 3$. The Coxeter matrix of $I_m$ is \begin{align} \left( \begin{matrix} 1 & m \\ m & 1 \end{matrix} \right). \end{align} When $m=3,4,6$, $I_m$ …

Are there some references which proves the following result? Let $W$ be a Coxeter group and $w \in W$. Then different reduced expressions of $w$ can be transformed from one into anther using only the …

In the paper, a set $L$ associated to an element $w$ in a Coxeter group $W$ is defined as follows. Let $w=s_{i_1} \cdots s_{i_m}$ be a reduced expression. Define $L=\{\beta_1, \ldots, \beta_m\}$, wher …