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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617 views

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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250 views

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 $S_n$ ...
4
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2answers
818 views

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, \begin{equation} H(n,L)\colon=\sum_{Y_{i,j,w}} ...
13
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0answers
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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 ...
4
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1answer
629 views

some confusion about the explicit construction of irreducible representations of $S_n$

In this book chapter, the irreducible representations of the symmetric group $S_n$ is given in terms of polytabloids of a Ferrer's diagram $\lambda$, defined as $e_t = \sum_{\pi \in C_t} ...
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4answers
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Making the branching rule for the symmetric group concrete

This question concerns the characteristic $0$ representation theory of the symmetric group $S_n$. I'm a topologist, not a representation theorist, so I apologize if I state it in an odd way. First, ...
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How ugly is the isomorphism R[GxH] = R[G] (X) R[H] for groups G, H?

Clearly, when $G$ and $H$ are two finite groups, and $V$ and $W$ are two representations of $G$ and $H$, respectively, then $V\otimes W$ is a representation of the group $G\times H$. It is a ...
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0answers
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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. ...
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1answer
513 views

Hankel determinants of symmetric functions

The starting point is that it is known that the Hankel determinants for the Catalan sequence give the number of nested sequences of Dyck paths. I would like to promote this to symmetric functions. ...
6
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1answer
840 views

schur Weyl duality in positive characteristic

Let $S_k$ be the symmetric group. Let $F$ be an algebraically closed field. Let $Rep(S_k)$ be the category of representations of $S_k$ over $F$. Let $Rep(GL_n(F))$ be the category of algebraic ...
3
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1answer
394 views

What is a concomitant (and other questions on D.E. Littlewood's “Products and plethysms of characters with orthogonal, symplectic, and symmetric groups” )?

I'm trying to understand the paper "Products and plethysms of characters with orthogonal, symplectic, and symmetric groups" by D.E. Littlewood (link), but I'm having trouble overcoming the language ...
12
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0answers
428 views

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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3answers
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Sarrus determinant rule: references, extensions

SEEKING REFERENCES FOR SARRUS' RULE AND EXTENSIONS An undergraduate came to me with an identity for 4x4 determinants that is actually correct: $\det(A)=h(A)+h(RA)+h(R^{2}A)$ where R cyclically ...
2
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2answers
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Algebraic structures at hypernatural parameters

Let's start with some family of algebraic structures of the same type indexed by the natural numbers, say the symmetric group $S_n$. Suppose that the axioms of this algebraic structure (in this case, ...
10
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1answer
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Reference request: The stable Kronecker ring is isomorphic to the ring of symmetric polynomials

Background For $\lambda$ any partition and $n$ a positive integer, write $\lambda[n]$ for the sequence $(n - |\lambda|, \lambda_1, \lambda_2, \ldots, \lambda_r)$. For $n$ large enough, this is a ...
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1answer
513 views

Explicit computation of induced modules of semidirect products with the symmetric group

I've gotten stuck in a project I have been working on, essentially on the following combinatorial question about the symmetric group. One can obtain a 1-dimensional representation $M^n_c$ of the ...