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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A finite dimensional algebra associated to the symmetric group

Let $S_n$ be the finite group given as $n \times n$ permutation matrices. Define for a given field $K$ the algebra $B_n$ as the subalgebra of $M_n(K)$ generated by all permutation matrices of $S_n$. (...
Mare's user avatar
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2 votes
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Reference for the action of the Mullineux involution on a partition with an added good node

Could you help me to locate a reference in the literature to the following fact: if you act by the Mullineux involution $M$ on a partition $Q$ which is a union of a regular partition $P$ with an added ...
IntegrableSystemsEnthusiast's user avatar
8 votes
1 answer
251 views

Branching rule for Specht modules over Kazhdan-Lusztig basis

Suppose $\lambda\vdash n$ is a partition and $S^\lambda$ is the associated irreducible representation of $S_n$. As we know from the branching rule, we have an isomorphism of $S_{n-1}$ modules $$S^\...
GossipM's user avatar
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6 votes
1 answer
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Can Matsumoto's theorem for the symmetric group be proved using a monovariant?

This is a question that can be asked for any Coxeter group, but for the sake of simplicity I will restrict myself to symmetric groups. Recall the main definitions: Let $n$ be a nonnegative integer. ...
darij grinberg's user avatar
2 votes
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75 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 $|\gamma| + p w = n$. ...
Mark Wildon's user avatar
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6 votes
2 answers
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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 ...
Lagrenge's user avatar
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10 votes
1 answer
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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_{...
Mare's user avatar
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13 votes
0 answers
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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 ...
Richard Hepworth's user avatar
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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 ...
Turbo's user avatar
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3 votes
2 answers
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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 $n$ ...
gualterio's user avatar
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7 answers
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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_{...
John Baez's user avatar
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4 votes
1 answer
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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 ...
gualterio's user avatar
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7 votes
1 answer
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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}...
Jeanne Scott's user avatar
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2 votes
1 answer
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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{...
thedude's user avatar
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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 ...
Turbo's user avatar
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4 votes
1 answer
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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^{...
T. Amdeberhan's user avatar
3 votes
0 answers
107 views

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 ...
gualterio's user avatar
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1 vote
0 answers
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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 $\lambda^T$....
naughie's user avatar
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7 votes
1 answer
325 views

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 ...
M. Winter's user avatar
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10 votes
2 answers
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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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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 ...
gualterio's user avatar
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2 votes
1 answer
623 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 ...
Francesco Polizzi's user avatar
2 votes
0 answers
272 views

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 ...
David E Speyer's user avatar
3 votes
0 answers
157 views

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 ...
HIMANSHU's user avatar
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4 votes
1 answer
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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 ...
gualterio's user avatar
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8 votes
0 answers
294 views

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 ...
Abdelmalek Abdesselam's user avatar
7 votes
0 answers
181 views

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 ...
Mare's user avatar
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9 votes
1 answer
884 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 ...
Abdelmalek Abdesselam's user avatar
6 votes
1 answer
168 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 ...
Abdelmalek Abdesselam's user avatar
8 votes
0 answers
185 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(\...
Abdelmalek Abdesselam's user avatar
3 votes
1 answer
132 views

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^{}}}) ...
HIMANSHU's user avatar
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4 votes
0 answers
140 views

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 ...
Christoph Mark's user avatar
4 votes
0 answers
109 views

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 ...
Sepehrius's user avatar
  • 121
4 votes
1 answer
274 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 ...
Christoph Mark's user avatar
7 votes
0 answers
118 views

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 ...
Sepehrius's user avatar
  • 121
3 votes
0 answers
127 views

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: '...
S.Farr's user avatar
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0 answers
379 views

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) $$\...
Stefan Witzel's user avatar
3 votes
0 answers
61 views

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 ...
sleeve chen's user avatar
4 votes
1 answer
268 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: ...
sleeve chen's user avatar
10 votes
1 answer
346 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 ...
user avatar
9 votes
2 answers
515 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 $x_1,x_2,...,x_n$ ...
Karthik C's user avatar
  • 261
3 votes
1 answer
194 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) ...
A. Gupta's user avatar
  • 336
4 votes
2 answers
430 views

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)...
vidyarthi's user avatar
  • 2,007
2 votes
1 answer
246 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 ...
Bipolar Minds's user avatar
2 votes
0 answers
45 views

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 ...
Chris Bowman's user avatar
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3 votes
0 answers
73 views

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 ...
eti902's user avatar
  • 835
-1 votes
1 answer
94 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 ...
dohmatob's user avatar
  • 6,706
3 votes
0 answers
325 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 \...
Volodymyr Chelnokov's user avatar
1 vote
0 answers
177 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,\...
GGT's user avatar
  • 685
1 vote
0 answers
76 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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