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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Not very transitive actions

Suppose $m$ is a positive integer. I am looking for finite sets with group actions such that the action is transitive on the set of $m$-element subsets, but NOT transitive on the set of $(m+1)$-...
Anton Petrunin's user avatar
2 votes
1 answer
299 views

Evaluations of group characters on cosets of subgroups

Let $G$ be a finite group, $H$ a subgroup of $G$ and $g \in G$. Define $$ [gH] = \sum_{h \in H} gh, $$ viewed an element in the group algebra $\mathbb{C}[G]$. Given an irreducible character $\chi$ of $...
Zach H's user avatar
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2 votes
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Symmetric polynomial constructed from symmetric group

Let $n$ be a positive integer, $S_n$ be the symmetric group. For a permutation $p=[p_1,\dots,p_n]\in S_n$, define $x^p := x_1^{p_1}\cdots x_n^{p_n}$. It can be seen that the following polynomial is ...
Max Alekseyev's user avatar
0 votes
0 answers
71 views

Integral of elements of random unitaries

It is known how to calculate the integral of elements of $N\times N$ Haar random unitaries using the Weingarten function: $$\int \prod_{k=1}^n U_{i_kj_k} U_{m_kr_k}^* \mathrm d U = \sum_{\sigma,\tau} \...
user50394's user avatar
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3 votes
0 answers
124 views

Filtrations of the irreducible representations of the symmetric groups

For a partition $\lambda$ of $n$, write $S^{\lambda}$ for the irreducible $S_n$-representation that corresponds to $\lambda$ (a.k.a. the Specht module). For two integers $d<n$ write $Par_d(n) = \{\...
Ehud Meir's user avatar
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7 votes
1 answer
294 views

What is the Lie superalgebra generated by permutations?

Consider the group algebra of the symmetric group $\mathbb{C}S_n$. Then there is a corresponding Lie algebra $\mathfrak{L}(S_n)$ defined by $$[\sigma, \tau] = \sigma\circ\tau - \tau\circ\sigma,$$ ...
WunderNatur's user avatar
0 votes
1 answer
291 views

Lower bound of the largest irreducible character degree of alternating group $A_n$

$\newcommand\cd{\mathrm{cd}}$Let $A_m$ and $A_n$ be two alternating groups and $15\le m+2 \le n$. Denote $\cd_m$ and $\cd_n$ as the largest irreducible character degree of $A_m$ and $A_n$, ...
Sun's user avatar
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15 votes
1 answer
566 views

What is the centralizer of a Young subgroup of $S_n$?

In their celebrated paper "A new approach to the representation theory of the symmetric group. II", Okounkov and Vershik prove that $Z(n-1,1)$, the centralizer of $\mathbb{C}[S_{n-1}]$ in $\...
Alvaro Martinez's user avatar
7 votes
2 answers
394 views

3-coloring the alternating group graph

Consider the alternating group graph, here defined as a Cayley graph on the alternating group $A_n$ using the generating set $\{(1,2,3),(1,2,4),\dotsc,(1,2,n),(1,n,2),\dotsc,(1,4,2),(1,3,2)\}$. Note ...
vidyarthi's user avatar
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6 votes
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What about Hopf algebra and fusion structures for intertwiner algebras?

Let $G$ be a complex, reductive group and let $V_1, \dotsc, V_r$ be a collection of finite dimensional, irreducible complex representations of $G$. Let $\mathcal{A} = \mathrm{End}_G(V_1 \otimes \dotsb ...
Jeanne Scott's user avatar
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3 votes
0 answers
104 views

Maximal generalized symmetric groups and the tensor product

Let $S(m,n)$ be the generalized symmetric group which is a wreath product of the cyclic group of order $m$, denoted here by $\mathbb{Z}_m$, and the symmetric group $S_n$. A standard unitary ...
Jonas Anderson's user avatar
3 votes
1 answer
345 views

What is the unit group of the representation ring of the symmetric group S_3?

The representation ring of the symmetric group S_3 is a fusion ring of rank 3. We wonder its unit group. By direct computation, it needs to solve two inhomogeneous Diophantine equations, but which we ...
Yunnan Li's user avatar
5 votes
1 answer
253 views

On a proof involving Young symmetrizers acting on tensor spaces

I hope this is not too elementary for this site, but I already asked something similar on MSE which has not received any attention whatsoever. I am extremely unfamiliar with the algebraic/...
Bence Racskó's user avatar
1 vote
1 answer
191 views

Words representations of elements of a symmetric group

Let $S=\{(1,2),(1,2,\ldots,n),(1,n,,n-1,\ldots,2)\}$ be a subset of the symmetric group $S_n$. Let $a=(1,2),b=(1,2,\ldots,n),c=(1,n,n-1\ldots,2)$ be the elements of $S$. My question is, since $S$ is a ...
vidyarthi's user avatar
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5 votes
0 answers
119 views

Towers of algebras, their 2-step centralizer algebras, and analogues of the degenerate affine Hecke algebra

Let $\, \big( {\frak{F}}_0 \subset {\frak{F}}_1 \subset {\frak{F}}_2 \subset \cdots \big)$ be a tower of semi-simple, finite dimensional, unital, complex algebras starting with ${\frak{F}}_0 \cong \...
Jeanne Scott's user avatar
  • 1,847
2 votes
0 answers
100 views

Partial sum of Weingarten functions over symmetric group

I have a question about partial sums of Weingarten functions. The Weingarten functions are defined as $$ E_U[U_{i_1,j_1}\dotsm U_{i_k,j_k}U^*_{i'_1,j'_1}\dotsm U^*_{i'_k,j'_k}]=\sum_{\alpha,\beta \in \...
postasguest's user avatar
1 vote
0 answers
98 views

Are there standard short notations for ascending and descending cyclic permutations?

In a paper I am currently writing I use cyclic permutations of the form $$ (k,k+1,\dots,\ell) $$ and $$ (\ell,\ell-1,\dots,k) $$ of consecutive elements quite a lot (I added the commas to avoid ...
M.G.'s user avatar
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8 votes
1 answer
348 views

Harmonic flow on the Young lattice

Let me begin with some preliminary concepts: A positive real-valued function $\varphi: P \rightarrow \Bbb{R}_{>0}$ on a locally finite, ranked poset $(P, \trianglelefteq)$ is harmonic if $\varphi(\...
Jeanne Scott's user avatar
  • 1,847
6 votes
1 answer
472 views

Is there a classification of homomorphisms $S_n \to S_{n+k}$ for small $k$?

Homomorphisms $B_n \to B_{2n}$ and $B_n \to S_{2n}$ have been classified in Chen–Kordek–Margalit - Homomorphisms between braid groups and Lin - Braids and permutations respectively. I am interested in ...
Noah Caplinger's user avatar
17 votes
0 answers
343 views

Row of the character table of symmetric group with most negative entries

The row of the character table of $S_n$ corresponding to the trivial representation has all entries positive, and by orthogonality clearly it is the only one like this. Is it true that for $n\gg 0$, ...
Sam Hopkins's user avatar
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6 votes
0 answers
222 views

Proving the spectrum of the Young-Jucys-Murphy elements by formal computation in the degenerate affine Hecke algebra

This is really a followup to Why are Jucys-Murphy elements' eigenvalues whole numbers? , specifically to Igor Makhlin's beautiful answer. I'm trying to make it even more beautiful by getting rid ...
darij grinberg's user avatar
1 vote
0 answers
75 views

Determine whether a set generates a residue field of an invariant ring

Fix two positive integers $m>n$. Let $(A|Y)$ be an $m\times (n+1)$ augmented matrix consisted of $m\times (n+1)$ indeterminates, where $Y$ is a column symbolic vector of length $m$. Denote $R=\...
GiS's user avatar
  • 321
2 votes
1 answer
193 views

Sequence of monotone tuples and permutation condition for rotation

I was doing some counting in $S_n$ symmetric group I encountered the following problem, which also someway related to central factorial number. So given a $n$ cycle say $(1,2,\ldots,n)$, what are the ...
GGT's user avatar
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2 votes
0 answers
73 views

Confusion Regarding Character Polynomials and Dimensions of Irreducible Representations in the Symmetric Group

I am trying to use the Garsia–Goupil formula. Fundamentally, the character polynomial satisfies $$ \chi^{(n-|\mu|, \mu)}_{1^{a_1} 2^{a_2} \cdots} = q_\mu(a_1, a_2, \ldots) \equiv q_\mu(1^{a_1} 2^{a_2} ...
Sam OT's user avatar
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10 votes
0 answers
378 views

Has anyone met this "$q$-character" table for $S_4$?

Is anyone aware of the following $q$-character table for the symmetric group $S_4$? \begin{array}{|c|c|c|c|c|c|} \hline \mathrm{conj}\backslash\mathrm{rep} & 2+1+1 & 3+1 & ...
Jeanne Scott's user avatar
  • 1,847
10 votes
2 answers
571 views

Littlewood-Richardson coefficients in terms of Specht modules

Littlewood-Richardson coefficients $c_{\nu\mu}^{\lambda}$ (where $\nu$,$\mu$ and $\lambda$ are integer partitions such that $|\nu| + |\mu| = |\lambda|$) are well-known coefficients appearing in ...
eti902's user avatar
  • 835
4 votes
1 answer
230 views

Relation between two permutation metrics

Note: I asked this question a few months ago here, but received no answer. Consider the following two metrics on permutations of $\{1,2,\dots,n\}$: $d_\text{swap}(\sigma,\tau)$ is the minimum number ...
reservoir's user avatar
8 votes
3 answers
1k views

Bijective proof for a partition identity

I came across the following cute fact about partitions: \begin{align} & |\{\lambda \vdash n \text{ with an even number of even parts}\}| \\[8pt] & {} - |\{ \lambda \vdash n \text{ with an odd ...
Nate's user avatar
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6 votes
2 answers
314 views

Is there a combinatorial interpretation of this array in terms of $S_{2n+1}$?

I have recently encountered a triangular array $(a_{i,j})_{0\le i\le j}$, each line of which might (should?) have a combinatorial interpretation in terms of $S_{2n+1}$. Here it is (the first entry of ...
Wolfgang's user avatar
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3 votes
2 answers
213 views

Swaps in a permutation across an index

We are given two positive integers $N$ and $K$ such that $K < N$. We start with an array $A=[1,2,\dots,N]$. We can choose an arbitrary index $i \in \{1,2,\dots,N-1\}$ and we can swap $A[i]$ with $A[...
David Pal's user avatar
15 votes
1 answer
501 views

Branching rule of $S_n$ and Springer theory

Let $u\in\mathrm{GL}_n$ be a unipotent element, let $\mathcal{B}_u$ be the variety of Borel subgroups containing $u$, and let $d=\dim \mathcal{B}_u$. Then Springer theory tells us that $H^{2d}(\...
user148212's user avatar
  • 1,534
6 votes
0 answers
235 views

What do Macdonald polynomials hint about $\operatorname{Rep}(S_\infty)$?

$\DeclareMathOperator\Rep{Rep}$It is well-known that the Macdonald "$P$" polynomials deform the Jack "$J$" polynomials [1]. The latter have profound relations with representation ...
Student's user avatar
  • 5,008
6 votes
1 answer
531 views

Representation ring of the symmetric group $S_n$ in the limit as $n \to \infty$

Let $S_n$ denote the symmetric group on $n$-letters and let $\mathrm{Rep}(S_n)$ denote its representation ring. For every $n$ restriction along the inclusion $S_{n-1} \to S_n$ induces a ring ...
Saal Hardali's user avatar
  • 7,549
12 votes
2 answers
1k views

The character table of the symmetric group modulo m

Let $S_n$ be the symmetric group and $M_n$ the character table of $S_n$ as a matrix (in some order) for $n \geq 2$. Question: Is it true that the rank of $M_n$ as a matrix modulo $m$ for $m \geq 2$ ...
Mare's user avatar
  • 25.7k
2 votes
0 answers
124 views

invariant decomposition of $\mathbb{C}[S_k^n]^{S_k}$

Denote $S_k^n = \underbrace{S_k \times \dots \times S_k}_{n \text{ times}}$ and let $S_k$ act on $S_k^n$ conjugate diagonal, so that $$ \pi (\sigma_1, \dots, \sigma_n)\pi^{-1} := (\pi \sigma_1 \pi^{-1}...
Felix Huber's user avatar
4 votes
1 answer
130 views

The symmetric group of positive integers acting on $\ell^\infty(\mathbb{N})^*$

Let $\mathfrak{S}_\mathbb{N}$ be the symmetric group of all positive integers. Let $\ell^\infty(\mathbb{N})^*$ be the dual space of $\ell^\infty(\mathbb{N})$ equipped with weak*-topology. There is a ...
Muduri's user avatar
  • 125
1 vote
0 answers
71 views

Scalars by which symmetrizations of cyclic permutations act on Specht modules

Let $S_n$ be the symmetric group. Pick $a \in 2,\ldots,n$ and denote by $c_a \in \mathbb{C}[S_n]$ the symmetrization of the element $(12\ldots a)$ i.e. $c_a$ is the sum of cycles of type $a$. Let $\...
Asav's user avatar
  • 153
1 vote
0 answers
80 views

How to obtain explicit formula for this sum over Young diagram?

Consider the next essence $$ B_N (r, q) =\sum_{\tau \vdash r} d^2 (\tau ) \prod_{i = 1}^{r} \frac{\Gamma [N + \tau_i - i +1]}{\Gamma [N + \tau_i - i +1+q]} $$ where $d(\tau)$ is dimension of ...
Sergii Voloshyn's user avatar
2 votes
1 answer
132 views

How to re-expand the sum of Schur function?

Consider next sum \begin{eqnarray} \label{PF_spindef} Z = \sum_{r=0}^{N N_f} h^{2r} \ Q(r) . \end{eqnarray} and \begin{equation} Q(r) \ = \ \sum_{\sigma \vdash r} s_{\sigma}(1^{N_f}) \ s_{\sigma}...
Sergii Voloshyn's user avatar
2 votes
0 answers
164 views

New identity for sum over Young diagram of symmetric group?

Consider the next identity $$ \sum_{\tau \vdash r} d^2 (\tau ) \! \prod_{i = -(r-1)}^{r-1} \! \! \left( N+i \right)^{t_i^r} =\sum_{\tau \vdash r} d^2 (\tau ) \prod_{i = 1}^{r} \Gamma [N + \tau_i - ...
Sergii Voloshyn's user avatar
22 votes
6 answers
2k views

What is the standard 2-generating set of the symmetric group good for?

I apologize for this question which is obviously not research-level. I've been teaching to master students the standard generating sets of the symmetric and alternating groups and I wasn't able to ...
Matthieu Romagny's user avatar
2 votes
0 answers
194 views

Geometric or combinatorial interpretations of the (weak) Bruhat order?

$\DeclareMathOperator\Inv{Inv}$The weak Bruhat order on the symmetric group has a straightforward combinatorial interpretation: Consider a set of labelled balls $1,2,\dotsc,n$. Then for two ...
Brendan Mallery's user avatar
7 votes
0 answers
180 views

Explaining $\left(a-1\right)^n \cdot n! \mid a^{n-1} \prod_{i=1}^n \left(a^i-1\right)$ by a free $S_n$-action

Here is an olympiad-level problem on elementary number theory: Let $a$ be an integer and $n$ a positive integer. Prove that \begin{align} \left(a-1\right)^n \cdot n! \mid a^{n-1} \prod_{i=1}^n \left(...
darij grinberg's user avatar
5 votes
0 answers
76 views

Finite trees with forests realizing all partitions

Removing interiors of some edges in a tree with $n$ vertices leaves a spanning-forest with $k$ connected components (given by subtrees) having respectively $\lambda_1,\ldots,\lambda_k$ vertices. We ...
Roland Bacher's user avatar
3 votes
0 answers
124 views

Ext groups of tensor products of Specht modules

$\newcommand{\sym}{\mathfrak{S}}$ $\DeclareMathOperator{\Ext}{Ext}$ Let $k$ be a field of characteristic $p > 0$ and $a < p \leq b$ so that the $k\sym_a$-modules are semisimple but $k\sym_b$-...
Cihan's user avatar
  • 1,586
3 votes
1 answer
96 views

Does a finite coresolution with Specht-filtered modules imply a Specht filtration?

$\newcommand{\sym}{\mathfrak{S}}$ $\newcommand{\rarr}{\rightarrow}$ Given a partition $\lambda$ of $n$ and a commutative ring $R$, writing $\sym_n$ for the symmetric group, there is a Specht module $S^...
Cihan's user avatar
  • 1,586
4 votes
1 answer
211 views

A transitive action on a specific set

Let $G$ be a finite group and $\lambda\in G$, consider a set $$D^{p+*}_{G}(\lambda):=\{P\in S^{p+*}_G|P^{\lambda}=P=[P,\lambda]\}$$ where: $S^{p+*}_{G}$ denotes the set consisting of all non-trivial $...
Ling's user avatar
  • 311
3 votes
1 answer
236 views

Probability that k randomly drawn permutations can be arranged to compose to the identity

Consider the symmetric group $S_n$ under the uniform distribution. For integer $k > 1$, suppose we draw $k$ elements $s_1, \dots, s_k$ independently at random. What is the probability that there ...
Nate River's user avatar
  • 4,802
3 votes
1 answer
248 views

Is the Cayley distance on permutation (matrices) equivalent to the Riemannian metric on $O(n)$?

Denote by $d_C(\sigma,\mu)$ the minimal number of transpositions needed to go from a permutation $\sigma$ to a permutation $\mu$. E.g. if $d_C(\sigma,\mu)=0$, then $\sigma=\mu$, if $d_C(\sigma,\mu)=1$,...
Peter's user avatar
  • 131
2 votes
0 answers
129 views

Braid 2-groups, symmetric 2-groups

Is there an object which can be called a "braid 2-group"? Or a "symmetric 2-group"? (Note: not a braided 2-group) I am ignorant about 2-categories but I hope that a good candidate ...
Alex Ogg's user avatar
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