Questions tagged [symmetric-groups]

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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0answers
40 views

Upper bound on decomposition numbers for the symmetric group in a block of weight $w$

The $p$-blocks of the symmetric group $S_n$ are labelled by pairs $(\gamma, w)$ where $\gamma$ is a $p$-core partition, $w \in \mathbb{N}_0$ is the weight of the block, and $|\gamma| + p w = n$. ...
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Involutive solutions to the Yang-Baxter equation (and triangular Hopf algebras)

I'm interested in solutions to the Yang-Baxter equation $$R_{12}R_{23}R_{12}=R_{23}R_{12}R_{23},$$ that are involutive $R^2_{12}=1$. Or put it another way, I'm interested in representations of the ...
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1answer
249 views

When are immanants irreducible?

For a partition $\lambda$ let $\chi_\lambda$ be the corresponding irreducible representation of the symmetric group $S_n$. Let $\mathrm{Imm}_\lambda(x) = \sum\limits_{\pi \in S_n} \chi_\lambda(\pi) x_{...
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Hidden grading on $kS_n$

Brundan and Kleshchev showed that if $k$ is a field of characteristic $p$, then the group ring $kS_n$ of the symmetric group $S_n$ admits an integer grading which is nontrivial in the sense that it is ...
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Greatest common length of permutation

Given two permutations $\pi_1$ and $\pi_2$ without their cycle decompositions is there a good method to compute the largest cycle length common between them in their decompositions? a good method to ...
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2answers
238 views

Is there any simple formula for the character of $S_{n}$ represented by the set of $k$-tuples of $\{1,2,…,n\}$?

I'm interested in the representation theory of symmetric groups. I'm now trying to search for the formula for the characters of $\Omega^{k}$, the set of $k$-tuple of elements of $\Omega$ a set of $n$ ...
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Representations of products of symmetric groups

I'm writing a paper and want to cite some references to efficiently prove that over any field $k$ of characteristic zero, every irreducible representation of a product of symmetric groups, say $$ S_{...
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1answer
94 views

What is the $p$-regular partition corresponding to the sign representation of $S_{n}$ over a field of characteristic $p$?

I'm now interested in the modular representation of symmetric groups. It is well-known that for a fixed prime $p$, there is a bijection between the irreducible representations of $S_{n}$ over a field ...
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1answer
422 views

Hurwitz numbers and $t$-cores

For integers $k \geq 0$ and $d \geq 1$ let $H(k,d)$ be the Hurwitz number which, for the purposes of this posting, will be defined by: \begin{equation} H(k,d) \, := \ d! \, \sum_{\lambda \, \vdash d}...
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Changing $S_2 \wr S_n$ for $S_n \wr S_2$ in the theory of zonal polynomials

The permutation group $S_{2n}$ has $H_{2,n}=S_2\wr S_n$ as a subgroup. The plethysm $h_n(h_2)=\sum_{\lambda\vdash n}s_{2\lambda}$ is well known. The zonal spherical functions $\omega_\lambda(g)=\frac{...
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On the real and finite field rank of a $0/1$ matrix - II

Let $M\in\{-\ell,\dots,-1,0,+1,\dots,+\ell\}^{n\times n}$ be a matrix of rank $r$ where $\ell\geq1$ such that there is a permutation matrix in $\{0,1\}^{m\times m}$ of order $2\ell$. Fix a permutation ...
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1answer
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A close reative of “Inflated” Eulerian polynomials

I came across this post Coefficients of the Inflated Eulerian Polynomial by AULI-GRAHAM-SAVAGE. In particular, the polynomials related to descents interested me $$P_n(x)=\sum_{\pi\in\mathfrak{S}_n}x^{...
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Reference for the Netto's theorem on the permutation groups which was mentioned in the paper of Frobenius

I'm trying to read 'Uber die Charaktere der mehrfach transitiven Gruppen' written by Frobenius. There he mentioned some theorems of Netto. I'm depending on the Google translator. and the translation ...
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Irreducible representation of transposed Young diagram of $\mathfrak{S}_n$ [closed]

For a Young diagram $\lambda$, let $V_\lambda$ be an irreducible representation of $\mathfrak{S}_n$ corresponding to $\lambda$ (over $\mathbb{C}$). And denote the transpose of $\lambda$ by $\lambda^T$....
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1answer
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For which $n$ can $S_n$ act transitively on $n+k$ elements?

It is known that the symmetric group $S_n$ can act transitively on $n+1$ elements if and only if $n=5$. Are there similar classifications for $S_n$ acting transitively on $n+k$ elements, where $k$ is ...
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2answers
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Arbitrarily large finite irreducible matrix groups in odd dimension?

I consider a finite irreducible matrix group $\Gamma\subseteq\mathrm{GL}(\Bbb R^d)$. I am interested in the maximal size of $\Gamma$ depending on $d$. But this question makes only sense if there is an ...
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186 views

Is there any research on the action of a subgroup on the whole finite group by conjugation?

I want to know whether there are any research on the orbits of the action of a subgroup by conjugation on the whole group, when the group is finite. (Especially whole symmetric group.) I'm especially ...
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1answer
582 views

Combinatorial problem in $\mathsf{S}_4$

I am working on a problem in Combinatorial Group Theory related to a construction in Algebraic Geometry, and I would like to have a conceptual proof of the fact described below. I am looking for ...
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Finding the decorated permutation of a non-reduced plabic graph

This is a question about Postnikov's theory of positroids and plabic graphs. The short version is If we have an non-reduced plabic graph $G$, how do we look at the alternating strands and read off ...
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Faithful representation into $\operatorname{GL}(9,3)$

Take $T=\big(\left< (123) \right> \times \left< (456) \right> \times \left< (789) \right>\big) \rtimes \left< (147)(258)(369) \right> \leq S_9$. Does there exist an injective ...
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1answer
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I want to know the name of or any references for a matrix in the book “The representation theory of the symmetric groups” by Gordon James

$\DeclareMathOperator{\Ind}{\operatorname{Ind}}$I'm reading "The representation theory of the symmetric groups" written by Gordon James. I found the matrix $B$ in the chapter 6 ("The ...
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A question related to Young symmetrizers

Let $T$ be an arbitrary Young tableau (i.e., filling of the diagram of an integer partition $\lambda$ of $n$ by the numbers from $1$ to $n$, each appearing once). Let $R(T)$ be the subgroup of ...
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Classification of posets that are quotient posets of the Boolean lattice

Quotient posets of the Boolean lattice $B_n$ have interesting properties and are for example discussed in chapter 5 of Stanley's book on algebraic combinatorics. $B_n/G$ for a subgroup $G$ of the ...
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1answer
237 views

A basic question about Young symmetrizers

This is probably elementary for experts on the representation theory of the symmetric group, but I did not find the answers I need by a cursory look at the usual textbooks (they could be there, but I ...
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1answer
141 views

On a certain expansion in term of Schur functions

This question is related to this other one A Schur positivity conjecture related to row and column permutations by Richard Stanley (thanks to Sam Hopkins for letting me know about it). Consider a ...
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141 views

Generalized Young symmetrizers, why not?

For $n$ a positive integer, let $[n]=\{1,2,\ldots,n\}$. Consider two set partitions $\mathcal{A}=\{A_1,\ldots,A_p\}$ and $\mathcal{B}=\{B_1,\ldots,B_q\}$ of the set $[n]$. We will denote by $G(\...
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1answer
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Permutation representation of a finite $p$-group

In the (xi) group of the classification of groups of order $p^4$ given by W.Burnside in his book," Theory of Groups Of Finite Order". The group ($\mathbb{Z_{p^{2}}}\rtimes \mathbb{Z_{p^{}}}) ...
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Words that give rise to an enumeration of elements of the symmetric group

Let $\mathbb{S}_m$ be the symmetric group on $m$ letters. Let $n=m-1$. Let $\mathbf{w}=a_1\cdots a_r$ be a word on the alphabet $\{1,\ldots,n\}$. We say that $\mathbf{w}$ gives rise to an enumeration ...
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Question about generalizing Cauchy identity

One of the Cauchy identities says that $$\prod_{i,j}(1+x_iy_j) = \sum_\lambda s_\lambda (x_1, \cdots,x_m) s_{\lambda'} (y_1, \cdots,y_n) $$ Where $\lambda$ is a Young diagram, $\lambda'$ is the ...
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1answer
168 views

Number of paths in the Bruhat order in the symmetric group

Let $\mathbb{S}_m$ the symmetric group on $m$ letters. Let $v\in\mathbb{S}_m$, and consider paths in the Bruhat order like this: $1\lessdot v_1\lessdot\cdots\lessdot v$, where $\lessdot$ means the ...
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A question regarding an analog of Young symmetrizer: the product row and column preserving subgroups without sign representation

Consider a rectangular Young diagram $\lambda$ with $n = pq$ boxes, with $p$ rows and $q$ columns. If $C$ is the column preserving subgroup of $\lambda$ and $R$ is the row preserving subgroup, then we ...
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Consequences of Littlewood-Richardson rule

I am trying to read Deligne's paper 'Categories Tensorielles', and in the first chapter Deligne states some results obtained from the Littlewood-Richardson rule that I do not understand. He states: '...
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Character table of the symmetric group $S_n$ according to James

In James, "The representation theory of the symmetric groups" an algorithm is described to produce the character table of a symmetric group. The proof involves the equation (pp. 22,23) $$\...
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Suggested papers or reading for PDE (high dimension) reduction to ODE by symmetries

Could anyone please suggest related papers or article about the topic related to my one question below? Reduce PDE to ODE by dilation symmetry I also cite a paper in the link above. We know that ...
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1answer
143 views

Reduce PDE to ODE by dilation symmetry

I am reading Rodrigues, Henrion, and Cantwell - Symmetries and analytical solutions of the Hamilton–Jacobi–Bellman equation for a class of optimal control problems, p.753. Consider the following PDE: ...
8
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1answer
212 views

Induction step in Bóna and Ehrenborg's proof that the generating function of the alternating runs has -1 as a root of a certain multiplicity

This is a crosspost of a question I asked on Mathematics SE four months ago. Periodically bumping it and placing a bounty on it to attract more attention were to no avail. There are some comments ...
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2answers
314 views

Decomposing a polynomial ring into Specht Modules

Let $S_{\pi}$ where $\pi$ is an integer partition of $n$, denote the Specht module corresponding to $\pi$. I am trying to decompose the set of all homogeneous polynomials in $x_1,x_2,...,x_n$ ...
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1answer
140 views

action of symmetric group on the second exterior power

Let $e_i \wedge e_j \ (i < j)$ be a basis for the $\mathbb Z$-module $\wedge^2 \Gamma$, where $\Gamma = \mathbb Z^n$. Clearly $S_n$ acts on the module $\wedge^2 \Gamma$ via $$\pi(e_i \wedge e_j) ...
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2answers
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Transposition Cayley graphs are planar

Consider the Cayley graph $G$ with vertex set the elements of the symmetric group $S_n$ and generating set the set of minimal transposition generators of the group $S_n$, that is the set $S=\{(12),(13)...
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1answer
219 views

Recurrence relation for number of reduced words of longest element in $S_n$

Is there any recurrence relation known for the number of reduced words of the longest element in $S_n$ (not commutation classes)? Edit: Sorry for unaccepting the answer, but I realized that I really ...
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0answers
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Carter Payne homomorphisms and reduced expressions

Let $G$ be an algebraic group and $W$ denote the underlying affine Weyl group. I will label representations of the principal block of $G$ by their alcoves, which in turn I label by the corresponding ...
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0answers
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Expansion of polytabloids in the standard basis

would like to know the most efficient way to write a polytabloid in terms of standard ones. I know the Garnir elements, but using them to do calculations is hard. I also read about "quadratic ...
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1answer
81 views

On bounding a certain discrepancy between probability distributions on the symmetric group

Disclaimer. This is a follow up to a question I asked and answered on SE https://math.stackexchange.com/q/3579311/168758. The question was about upper-bounds. Here I'm interested in lower bounds, and ...
3
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0answers
205 views

Sum of products of irreducible characters of the symmetric group over a subgroup

When trying to build a dual formulation for lattice gauge theories using Weingarten integration I am getting sums of the kind $$I^{m, n}_{\mu, \nu} (\sigma, \tau) = \sum_{\pi \in S_n} \chi_\mu (\pi \...
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171 views

Combinatorial bijection on monotone sequences

Let $(n),\mu$ be the partition of $n$ define $H_g^{m}((n);\mu)$ count's the number of tuples $(\tau_1,\ldots,\tau_r)$ of transposition in symmetric group $S_n$ with the following conditions $$ (1,2,\...
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0answers
53 views

Singular chain complex of balanced products

Let $\pi\subseteq\Sigma_r$ and $V$ be a right $\pi$-space. We may assume that $V$ is free, if necessary. Consider the morphism of singular chain complexes (over a fixed commutative ring) $$f:C_*(V) \...
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2answers
292 views

What makes skew characters of the symmetric group special?

For integer partitions $\mu\subset\lambda$ we can define the skew character $\chi^{\lambda/\mu}$ (for example?) via the Littlewood-Richardson rule. Many combinatorial gadgets and algorithms extend in ...
7
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0answers
176 views

Non-zero group determinant for symmetric group

Let $G$ be a finite group. Given complex numbers $x=\{x_g: g\in G\}$, one can define a $|G|\times |G|$ matrix $X$, with entries $X_{g,h} = x_{gh^{-1}}$. Let's consider $G$ being the symmetric group $...
2
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0answers
68 views

CAS for finite-dimensional complex representations of $S_n$

Does there exist a computer algebra system that can work with finite-dimensional complex representations of the symmetric groups on finitely many letters? It should have the following functionality: (...
2
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1answer
116 views

Minimum local permutation data needed to globally merge locally sorted sequences?

We have $k$ blocks of integer sequences $B_1,\dots,B_k$ where $B_i$ is a sequence $$a_{i,1},\dots,a_{i,n_i}$$ with $a_{i,j}\leq a_{i,j+1}$. Denote the permutation matrix $M_{\ell,\ell'}$ that merges $...

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