# 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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### is the embedding $\mathrm{Sp}_{2m}(p)\leqslant S_{p^m-1}$ possible?

Is the following embedding possible?
$\mathrm{Sp}_{2m}(p)\leqslant S_{p^m-1}$ where $S_{p^m-1}$ is a symmetric group and $p$ is prime. I see that when $p=3$ and $m=3$, the order of the former does ...

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### Decomposition of a tensor product of representations of $\mathrm{GL}_l(\mathbb{C})$ and decomposition of Littlewood-Richardson numbers?

For a positive integer $m$, denote $T(m)=\{(\lambda_1,\dots,\lambda_m)\in \mathbb{Z}^m:\lambda_1\ge \lambda_2\ge\dots \ge\lambda_m\}$ and $T^+(m)=\{ (\lambda_1,\dots,\lambda_m)\in \mathbb{Z}^m:\...

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### Hamiltonian edge colouring of complete graphs with even numbers of vertices

Edges of the complete graph on $2n$ vertices can be colored with $2n-1$ colors such that only edges of different colors intersect.
Can this always be done such that for every pair of different colors ...

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### Optimizing computations with nilpotents in a group algebra

Of course, I have a very concrete problem at hand, which has been vexing me for about a year now. But let me start with a question that has a better chance of having been answered.
Let $G$ be a ...

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### I search representation in terms of Schur Q-function

Consider next sum
$$
Z_0^{N, N_f} = \sum_{r=0}^{N N_f} \sum_{\lambda \vdash r }s_{\lambda}(1^{N_f})
s_{\lambda} (1^{N_f})
= \det_{1\le i, j \le N} \ \binom{2 N_f}{N_f-i +j} = s_{N^{N_f}} \left(...

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### How does Sage order the elements of the symmetric group?

In Sage, the symmetric group is a list. For instance if G = SymmetricGroup(3), we have
\begin{align*}
G[0] & = e \\
G[1] & = (1,3,2)\\
G[2] & = (1,2,3) \\
G[3] &= (2,3)\\
G[4] &= (...

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### Proof of a combinatorial identity for a sum over partitions of sets giving rise to a symmetric polynomial?

Consider a set $N$ with elements $n_1, n_2, \dots, n_k$ which are distinct integers. Introduce the notation $N_{i=1,2,\dots,s}$ for the $s$ blocks of a set partition of $N$. Consider a supplementary ...

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### Are there invariants of configurations of points in space obtainable via the moduli space of solutions of the Berry-Robbins problem?

Let $C_n(\mathbb{R}^3)$ denote the configuration space of $n$ distinct points in Euclidean $3$-space and let $U(n)/T^n$ denote the flag manifold associated to the unitary group $U(n)$, i.e. the ...

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### A probability problem in the conjugacy classes of symmetric group

Assume that $\sigma\in S_n$ has the cycle type $(p,.,p,1,..,1)$ where $p>2$ is a prime and the numbers of $1$ maybe $0$. If $\sigma_1$ and $\sigma_2$ are chosen uniformly in the conjugacy class of $...

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### Formula for the off-diagonal "elementary matrices" in the Gelfand-Tsetlin basis of the symmetric group algebra?

Are there any formulas for the irreducible off-diagonal elements $E^{\lambda}_{ij}$ in the Gelfand-Tsetlin basis of the symmetric group algebra $\mathbb{C}[S_n]$?
Here is the context for my question. ...

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### Large values of characters of the symmetric group

For $g$ an element of a group and $\chi$ an irreducible character, there are two easy bounds for the character value $\chi(g)$: First, the bound $|\chi(g)|\leq \chi(1)$ by the dimension of the ...

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### Clebsch–Gordan decomposition formula for algebraic groups

$\DeclareMathOperator\SL{SL}$There is a well-known Clebsch–Gordan decomposition formula for irreducible representations of $\SL_2$. If $V_n$ denotes the unique $n+1$-dimensional irreducible ...

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### Alternate proof in Fulton–Harris of representation theoretic version of Littlewood–Richardson rule

$\DeclareMathOperator\Ind{Ind}$Let $d = d_1 + d_2$ with $d_1$, $d_2$ positive integers. Let $\lambda$ be a partition of $d_1$ and $\mu$ a partition of $d_2$, so that the Young symmetrizer construction ...

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### Asymptotic character theory of unitary groups via shifted Schur functions

In the paper "Shifted Schur Functions" http://arxiv.org/abs/q-alg/9605042 by Andrei Okounkov and Grigori Olshanski it is said that one of the motivations for that paper was the asymptotic ...

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### Characters of alternating groups

I am studying certain properties of characters of alternating groups and I have found precisely three characters not satisfying it up to $A_{15}$. These are:
A character of dimension $3.696$ of $A_{...

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### Using Schur-Weyl duality

I am trying to gain a better understanding of Schur-Weyl duality specifically applied to symmetric functions. My motivating example is trying to understand the Frobenius character of the multilinear ...

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### Some combinatorics question concerning symmetric groups

Let $n = ht$ where $n, h ,t $ are all positive integers. I want to count $\omega \in S_t$ satisfying the following two properties:
$\omega(t+1 - \omega(i)) = t+1 - i$.
$\sum_{i: i \geq \omega(i)} (h ...

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### Jucys-Murphy elements and permutation modules

So I just learn about Jucys-Murphy elements. They are elements of $\mathbb{C}[\mathfrak{S}_n]$, the group algebra of the symmetric group, defined as:
$$
X_i = \displaystyle \sum_{k=1}^{i-1} (k,i)
$$
...

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### Need for "minimal representation" of a symmetric group

I need to construct a representation of a symmetric group $S_n$, in which a character of the conjugacy class $(n)$ (a class of permutations, which are cycles of a maximal possible length $n$) would be ...

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### Detecting symmetries in polynomials that lead to nice geometric properties

If we plot the single variable polynomial $p(x) = (x^2-1)^2$, it is easy to see that it has a nice property: all of its local minima are actually global minima.
In particular, it has precisely two ...

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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 ...

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### 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 ...

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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 $(m+1)$-...

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### 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 $...

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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 ...

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### 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} \...

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### 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) = \{\...

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### 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
$$[\sigma, \tau] = \sigma\circ\tau - \tau\circ\sigma,$$
...

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### 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 $A_n$, ...

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### 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 $\...

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### 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 ...

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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 ...

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### 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 ...

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### 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 ...

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### 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/...

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### 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 ...

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### 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 \...

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### 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 \...

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### 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 ...

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### 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(\...

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### 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 ...

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### 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 $n\gg 0$, ...

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### 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 ...

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### 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=\...

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### 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 ...

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### 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} ...

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### 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 & ...

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### 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 ...

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### 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 ...

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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 ...