# 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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### Is there any good survey on the hook length formula and related topics?

I am recently doing some research related to the hook length formula. The hook formula counts the number of Young tableaux of certain type. I find there are plenty of research already been done and ...
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### Nekrasov-Okounkov hook length formula

I am now reading the paper An explicit expansion formula for the powers of the Euler Product in terms of partition hook lengths by Guo-Niu Han. The author rediscovered what he calls Nekrasov-Okounkov ...
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### Integral Cohomology of Symmetric Groups

Does anybody know a reference for the explicit description of the integral cohomology ring of $S_5$ and $S_6$. I can not find them anywhere in the internet. For $S_4$, I found C. B. Thomas's nice ...
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### research on the structure/properties of permutation matrix/table with $(i,j)th$ entry as $\pi_j\circ \pi_i^{-1}$

Is there any research on the structure/properties of permutation matrix/table with $(i,j)th$ entry as $\pi_j\circ \pi_i^{-1}$, where $\{\pi_1,\pi_2,...,\pi_{k!}\}=S_k$? I know if we apply the ...
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### Minimum word length for an unusual set of generators of the symmetric group

Problem. Let $n\geq 2$ and let $T$ be the set of all permutations in $S_n$ of the form $$t_k:=\prod_{1\leq i\leq k/2}(i,k-i) \qquad \hbox{for k=3,4,\ldots,n+1}.$$ Find the least integer $f_n$...
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Recall that a permutohedron is a graph on the set of permutations $S_n$ with an edge between $\sigma$ and $\tau$ if they differ by one adjacent transposition: $\tau = (i,i+1) \circ \sigma$ for some $i ... 2answers 445 views ### What is natural about the well-known bijection between conjugacy classes and irreps of a symmetric group? Symmetric groups possess a well-known bijection between conjugacy classes and irreducible representations. More precisely, both sets are indexed by Young diagrams. Question: To what extent is this ... 1answer 235 views ### A Simple Bijective Proof Of Stanley's Hook-Content Formula for Hook Shapes this is my first post on math overflow so I hope it goes well. I believe I have a fairly simple bijective proof for Stanley's Hook-Content Formula in the case of hook shapes. I wanted to see if ... 0answers 110 views ### Geometric interpretation of table with permutations and inversions Let$T(n,k)$is the number of permutations of numbers$1, ..., n$and each of the permutations has$k$inversions. We can consider a table for$T(n,k)$for some$n$and$k$. For eg.$n=1,...,6$,$k=1,....
There is a well-known construction of minimal idempotents in the group algebra of the symmetric group $\mathbb C[S_n]$ using row symmetrizers and column antisymmetrizers. But these idempotents are ...