# 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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### Sum of Young symmetrisers of a given shape

Preliminaries and notation: Let $n\in \mathbb{Z}_{>0}$ and $\lambda=(\lambda_1,\lambda_2,\dots,\lambda_s)\vdash n$ be a partition. Given a Young diagram of shape $\lambda$, we can associate it ...
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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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### 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 ...
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### Where do stable Kronecker coefficients live “in nature”?

Background: For a partition $\lambda$, let $\lambda[N] = (N - |\lambda|, \lambda_1, \lambda_2, \lambda_3, \dots)$, also let $\chi_\lambda$ be the corresponding irreducible character of the symmetric ...
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### What Is The Minimal Monomial of the Symmetric Group?

In the symmetric group $S_n$ what is the shortest sequence $c_1,\ldots,c_k\in S_n$ such that, for all $x\in S_n$ the following product of conjugates of $x$: $$x^{c_1}x^{c_2}\ldots x^{c_k}$$ equals the ...
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### A $q$-analogue of Foulkes' character related to alternating permutations

My paper "Alternating permutations and symmetric functions" at http://math.mit.edu/~rstan/papers/altenum.pdf enumerates certain classes of alternating permutations, such as those whose inverse is ...
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### Finding a maximal tableau for a sum of Jucys-Murphy Elements

Let $X \in \mathbb{C}[S_n]$ be an element of the group algebra of $S_n$ expressible as the sum of some Jucys-Murphy elements. Then let $\lambda$ be any irreducible representation of $S_n$, with the ...
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### Name for class of flattening permutations

Let $S_n$ be the symmetric group. For any sequence of numbers $y=[y_1,y_2,\cdots,y_k]$, define the flattening operation as $\mbox{flatt}_{k}(y)$ as a relabeling of $y_1,y_2,\cdots,y_k$ in terms of ...
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### Partial order on partitions and symmetric group algebra

Let $n$ be a natural number. Consider a set $\Lambda_n$ of partitions of $n$ into a sum of natural numbers, like $n = \lambda_1 + \cdots +\lambda_k$ (A set of small lambdas representing a partition is ...
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### Frobenius norms of Fourier coefficients of the symmetric group

Suppose $f \colon S_n \rightarrow \mathbb{R}$ is some weighted collection of permutations. We want to understand how "well-spread" $f$ is. Our first test is its actions on singletons - are the ...
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### Isometry from a representation to the representation tensored with itself

Suppose, the group $G=S(2^{\infty})$ has a unitary representation $\pi$ on a separable infinite dimensional Hilbert space $H$. (The group $S(2^{\infty})$ is the direct limit of the following ...
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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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### What groups of symmetry are most suited for filling uniformely a spherical 3D space, whilst possessing the lowest possible surface-to-volume ratio?

I am looking for the closest known approximate solution to Kelvin foams problem that would obey a spherical symmetry. One alternative way of formulating it: I am looking for an equivalent of Weaire–...
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### Necklaces and the generating function for inversions

The problem of Necklaces is well-known, i.e "The number of fixed necklaces of length $n$ composed of $a$ types of beads $N(n,a)$" can be calculated: http://mathworld.wolfram.com/Necklace.html Let us ...
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### Partitions limit shape and LDP

Hello! I am trying to understand the paper of Dembo-Vershik-Zeitouni, Large deviations for integer partitions. I am only interested in Theorem 2, which deals with the case of the uniform distribution....
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### Picking $n$ so that certain Schur functors of the standard representation of $S_n$ are linearly independent

Let $V_n$ be the standard permutation representation of the symmetric group $S_n$, and let $\mathbb{S}_{\lambda}$ denote the Schur functor associated to the partition $\lambda$. Let $\lambda$ range ...
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### Function recursion relation over symmetric group

Hi! Let P be a permutation in the symmetric group SN and let π=πj, j+1 be a transposition of elements j and j+1 of the permutation. Let A(P) be a function in dependence of the permutation P. P&...
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### Partial orders on tabloids

Let $n \in \mathbb{N}$ and let $\lambda \vdash n$, a partition of $n$. By a $\lambda$-tabloid I mean a row-tabloid of shape $\lambda$. There is a well-known order on the set of $\lambda$-tabloids, ...
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### What is the definition of plethysm in the representation theory of permutation groups

Let $s_\lambda \circ s_\mu$ be a plethysm. Here let $\lambda, \mu$ be $m,n$ box Young diagrams. I have seen the definition of plethysms in symmetric functions. I would like to understand the ...
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### Character sums over a fixed subset of skew tableaux

Let $f(\lambda)$ count the number standard young tableaux of shape $\lambda\vdash n$ and $\lambda=(\lambda_1,\cdots,\lambda_r)$. Let $\mu \vdash k$ be a partition for $k<n$. It is a consequence of ...
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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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### A question about divided differences

I want to ask a question about divided differences. Let $n\equiv0,1 \pmod 4$ is a positive integer. We know that for any polynomial $f\in \mathbb{Z}[x_1,x_2,\cdots,x_n]$, \partial_{w_0}(f)=\left(\...
### normal sets and conjugate generating sets of $S_n$
In this arXiv paper (p. 13), Steinhardt defines a normal set in $S_n$ as follows: Definition: A split set of more than two cycles generating $S_n$ is said to be normal if any element is adjacent to ...