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.
438
questions
8
votes
2
answers
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One element commutation classes of reduced decompositions of the longest element of the Weyl group
For the symmetric group on $n$ objects $S_n$ the question of how to write its longest element $w_0$ as a reduced decomposition is an important combinatorical problem. As example, in this question the ...
6
votes
1
answer
240
views
Length of $\mathbb{C}^\infty$ as an $S_\infty$-representation
We know that $0 \subseteq V_n \subseteq \mathbb{C}^n \cong \mathbb{1}_n \oplus V_n$ is a composition series for the natural $\mathbb{C}[S_n]$-module $\mathbb{C}^n$ for all $n \geq 2$.
Now we have ...
11
votes
2
answers
891
views
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 $(m+1)$-...
2
votes
1
answer
300
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 $...
2
votes
0
answers
91
views
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 ...
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:
$$\int \prod_{k=1}^n U_{i_kj_k} U_{m_kr_k}^* \mathrm d U = \sum_{\sigma,\tau} \...
3
votes
0
answers
124
views
Filtrations of the irreducible representations of the symmetric groups
For a partition $\lambda$ of $n$, write $S^{\lambda}$ for the irreducible $S_n$-representation that corresponds to $\lambda$ (a.k.a. the Specht module).
For two integers $d<n$ write $Par_d(n) = \{\...
7
votes
1
answer
294
views
What is the Lie superalgebra generated by permutations?
Consider the group algebra of the symmetric group $\mathbb{C}S_n$. Then there is a corresponding Lie algebra $\mathfrak{L}(S_n)$ defined by
$$[\sigma, \tau] = \sigma\circ\tau - \tau\circ\sigma,$$
...
0
votes
1
answer
292
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$, ...
15
votes
1
answer
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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 $\...
7
votes
2
answers
396
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 ...
6
votes
0
answers
138
views
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 ...
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 ...
3
votes
1
answer
347
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 ...
5
votes
1
answer
257
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/...
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 ...
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 \...
2
votes
0
answers
101
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 \...
1
vote
0
answers
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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 ...
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(\...
6
votes
1
answer
475
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 ...
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 $n\gg 0$, ...
6
votes
0
answers
223
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 ...
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=\...
2
votes
1
answer
195
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 ...
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} ...
10
votes
0
answers
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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 & ...
10
votes
2
answers
579
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 ...
4
votes
1
answer
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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 ...
8
votes
3
answers
1k
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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 ...
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 ...
3
votes
2
answers
216
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[...
15
votes
1
answer
502
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}(\...
6
votes
0
answers
236
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 ...
6
votes
1
answer
540
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 ...
12
votes
2
answers
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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$ ...
2
votes
0
answers
124
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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}...
4
votes
1
answer
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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 ...
1
vote
0
answers
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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 $\...
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 ...
2
votes
1
answer
134
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}...
2
votes
0
answers
165
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 - ...
22
votes
6
answers
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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 ...
2
votes
0
answers
197
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 ...
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(...
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 ...
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$-...
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^...
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 $...
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 ...