# Tagged Questions

The symmetric group $S_n$ is the group of permutations of the set of integers $\{1,\dots,n\}$. This has $n!$ elements and is generated by the $n-1$ involutions exchanging consecutive integers. The symmetric groups form the simplest family of Coxeter groups.

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### Double coset representatives and structure of hecke algebras

Let $GL_n(F_q)$ be the general linear group over finite field $F_q$ and $B_n$ be its borel subgroup consisting of all upper triangular matrices. Then the double cosets $B_n\backslash GL_n(F_q)/B_n$ ...
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### Can this nested sum be expressed in terms of generalized harmonic numbers and the cycle index polynomials of the symmetric groups?

For a paper I was working on recently I needed to find the value of the following sum: $$S(n,k) = \sum_{i_1 = 1}^n \sum_{i_2 = i_1+1}^n \cdots \sum_{i_k=i_{k-1}+1}^n \frac{1}{i_1 i_2 \cdots i_k}.$$ ...
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### Branching rule from symmetric group $S_{2n}$ to hyperoctahedral group $H_n$

Embed the hyperoctahedral group $H_n$ into the symmetric group $S_{2n}$ as the centralizer of the involution $(1, 2) (3, 4) \cdots (2n-1, 2n)$ (cycle notation). Label representations of $S_{2n}$ by ...
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### roots of permutations

Consider equation $x^2=x_0$ in symmetric group $S_n$, where $x_0\in S_n$ is fixed. Is it true that for each integer $n\geq 1$, the maximal number of solutions (square roots) has identity permutation? ...
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### A general formula for the number of conjugacy classes of $\mathbb{S}_n \times \mathbb{S}_n$ acted on by $\mathbb{S}_n$

$\def\S{\mathbb{S}}$ Dear all, So I have $\S_n$ acting on $\S_n \times \S_n$ via conjugacy. That is: for $g \in \S_n, (x,y) \in \S_n \times \S_n$: $g(x,y) = (gxg^{-1},gyg^{-1}).$ Is there a general ...
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### Decompositions for Symmetric Groups

I'm looking for information about how representations of $S_n$ decompose under restriction. I know about the branching rule: That is, in characteristic 0, irreducible modules $L(\lambda)$ for $S_n$ ...
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### A sum involving irreducible characters of the symmetric group

Recently, during the research, I came across a sum, denoted by $H(n,L)$, involving irreducible characters of the symmetric group, H(n,L)\colon=\sum_{Y_{i,j,w}} \frac{\chi^{Y_{i,j,w}([...
### Diagonalizing some matrices arising from Fourier transform on $S_n$.
Consider the function $f$ on $S_n$ which equals $1/n$ on all adjacent transpositions $(i,i+1)$, where we let $n+1 = 1$, and $0$ otherwise, and its Fourier transform $\hat{f}(\rho)$ evaluated at the ...