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.

Filter by
Sorted by
Tagged with
2 votes
0 answers
57 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 \...
user avatar
2 votes
0 answers
63 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 \...
user avatar
1 vote
0 answers
84 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 ...
user avatar
  • 5,661
7 votes
1 answer
218 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(\...
user avatar
6 votes
1 answer
311 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 ...
user avatar
16 votes
0 answers
294 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 $n\gg 0$, ...
user avatar
  • 18.8k
6 votes
0 answers
127 views

Noncommutative algebra question inspired by Young-Jucys-Murphy elements

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 ...
user avatar
1 vote
0 answers
73 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=\...
user avatar
  • 301
1 vote
1 answer
143 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 ...
user avatar
  • 621
2 votes
0 answers
60 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} ...
user avatar
  • 498
10 votes
0 answers
328 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 & ...
user avatar
9 votes
2 answers
294 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 ...
user avatar
  • 643
3 votes
1 answer
141 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 ...
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 ...
user avatar
  • 1,849
6 votes
2 answers
286 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 ...
user avatar
  • 12.4k
2 votes
2 answers
139 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[...
user avatar
12 votes
1 answer
314 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}(\...
user avatar
  • 1,399
6 votes
0 answers
200 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 ...
user avatar
  • 4,005
6 votes
1 answer
355 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 ...
user avatar
  • 6,927
11 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 $m \geq 2$ ...
user avatar
  • 21.8k
2 votes
0 answers
121 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}...
user avatar
4 votes
1 answer
110 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 ...
user avatar
  • 75
1 vote
0 answers
61 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 $\...
user avatar
  • 123
1 vote
0 answers
62 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 ...
user avatar
2 votes
1 answer
112 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}...
user avatar
2 votes
0 answers
131 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 - ...
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 ...
user avatar
2 votes
0 answers
82 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 ...
user avatar
7 votes
0 answers
172 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(...
user avatar
5 votes
0 answers
68 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 ...
user avatar
3 votes
0 answers
77 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 $k\sym_b$-...
user avatar
  • 1,266
3 votes
1 answer
77 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^...
user avatar
  • 1,266
4 votes
1 answer
192 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 $...
user avatar
  • 311
3 votes
1 answer
222 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 ...
user avatar
  • 1,149
3 votes
1 answer
169 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 $d_C(\sigma,\mu)=1$,...
user avatar
  • 131
2 votes
0 answers
120 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 ...
user avatar
  • 159
14 votes
4 answers
1k views

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 $S_n$. (...
user avatar
  • 21.8k
2 votes
1 answer
67 views

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 ...
user avatar
8 votes
1 answer
222 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^\...
user avatar
  • 221
5 votes
1 answer
239 views

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. ...
user avatar
2 votes
0 answers
50 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 $|\gamma| + p w = n$. ...
user avatar
  • 10.4k
5 votes
1 answer
154 views

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 ...
user avatar
  • 241
10 votes
1 answer
254 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_{...
user avatar
  • 21.8k
11 votes
0 answers
183 views

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 ...
user avatar
0 votes
0 answers
48 views

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 ...
user avatar
  • 12.8k
3 votes
2 answers
276 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 $n$ ...
user avatar
  • 951
10 votes
7 answers
1k views

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_{...
user avatar
  • 19.4k
4 votes
1 answer
192 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 ...
user avatar
  • 951
7 votes
1 answer
511 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}...
user avatar
2 votes
1 answer
179 views

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{...
user avatar
  • 1,033

1
2 3 4 5
8