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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 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 $V$ be a vector space. A reflection is a linear map $f: V \to V$ which has an eigenvalue $1$ with multiplicity $n-1$. Let $S_n$ be the symmetric group on $\{1,\ldots,n\}$. Then the reflections in …

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 …

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 …

Let $M$ be a monoid. If $ab=ac$ implies that $b=c$, $a,b,c \in M$, then $M$ is said to have the left cancellation property. Similarly, the right cancellation property is $ba=ca$ implies that $b=c$. …

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 …

Let $S_n$ be the symmetric group over $\{1,2,\ldots,n\}$. How to return elements of length $m$ in $S_n$ using Sage? I try to find such function in Sage but didn't find one. Thank you very much. Edit: …

By the answer of the question, the set of reflections of a Coxeter system $(W,S)$ is given by $R = \{ wsw^{-1} : w \in W, s \in S\}$. Do we have a one to one correspondence between positive roots and …

Let $S_n$ be the symmetric group on $\{1, \ldots, n\}$. Let \begin{align} T=\sum_{g\in S_n} g. \end{align} Are there some references about the factorization of $T$? In the case of $n=3$, we have \b …