# 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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### Matrix invariants for simultaneous conjugation by a finite subgroup of $\textrm{GL}_n$

Let $K$ be a field of characteristic 0, and consider $d$ generic $n\times n$ matrices $X_1,\ldots,X_d$ where $X_k = (x_{ijk})_{ij}$ and $ K[x_{ijk}]$ is the polynomial algebra in $n^2 \cdot d$ ...

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### Hyperoctahedral group, preliminaries [closed]

I am looking for information on the hyperoctahedral group
From what I understand, the hyperoctahedral group is 'the generalized symmetric group' in the case where $m=2$. That is, the hyperoctahedral ...

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### H^2 of symmetric group

I'm a number theorist in need of some group cohomology lemmas, and I'm rather bewildered by the level of generality used in the literature. Specifically, the result I need is as follows: the ...

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### How to multiply dots with Young idempotents in the degenerate affine Hecke algebra (type A)

Let $\widehat{\cal H}_n$ be the type A degenerate affine Hecke algebra on $n$ strands, and let $x_1,\cdots,x_n$ be the dots. Inside of this algebra lies the algebra $\mathbb C S_n$, and the Young ...

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### Characters with all higher exterior powers irreducible

Let $G$ be a finite group and we take for the field the complex numbers.
Call an irreducible character $\xi$ with degree $m$ of $G$ perfect, if all exterior powers $\bigwedge\nolimits^k \xi$ are ...

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### What is known about finite dimensional modules over the nilCoxeter algebra?

Recall that the nilCoxeter algebra $\mathcal{N}_W$ for a Coxeter group $W$ is given by the $\mathbf{k}$-basis $x_w$ for each $w\in W$ and multiplication $x_ux_v=x_{uv}$ if $\ell(uv)=\ell(u)+\ell(v)$ ...

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### Conjectures in the representation theory of the symmetric group

Question: What are current open conjectures about the representation theory of the symmetric group?
I am interested mostly in the characteristic 0 case, but conjectures for the modular case can also ...

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### Efficient decomposition algorithm for characters of symmetric groups

Let $\chi$ be a rational character of $G:=S_n$, and we want to know whether it decomposes into irreducibles $\chi_\lambda$, for $\lambda\in\Lambda$, with $\Lambda$ given, as
$$
\chi=\sum_{\lambda\in\...

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### The sum of the signs of conjugacy classes in the symmetric group S_n [duplicate]

Let $r$ be the number of conjugacy classes of the symmetric group $S_n$ whose sign is $1$, i.e.
\begin{equation}
r := \#\{c \in \text{Conj} (S_n): \text{sgn} (c) = 1 \}.
\end{equation}
Let $s$ be the ...

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### The meet of two dominant permutations in weak order of $S_n$

A permutation is called dominant if its Lehmer code is a partition, or equivalently if it avoids the pattern $132$.
I can prove that given a permutation $v\in S_n$, there is a unique dominant ...

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### Plethysm and wreath product

I am looking for a proof about the link between plethysm and wreath product. It is a well-known fact, being use extensively in many papers, but I can't find a good reference. Everything that follows ...

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### Using the Dold-Thom Theorem to define \'etale cohomology

For reasonable spaces $X$, the Dold-Thom Theorem states that $\pi_i(SP(X)) \cong \tilde{H}_i(X)$ where $SP(X) = \bigsqcup_i \mathrm{Sym}^i(X)$. There is a purely algebro-geometric realization of this ...

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### Relationship between the symmetric group representation (Specht module) of a Young diagram and the Young diagram obtained by deleting one row

Suppose $\lambda$ is a Young diagram, and $\lambda'$ is obtained by deleting one particular row of $\lambda$. Is there any relationship between the symmetric group representation (Specht module) ...

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### An identity for characters of the symmetric group

I am looking for a reference for the identity
$$\chi_\lambda(C)=\frac{\dim(V_\lambda)}{|C|}\sum_{p\in P_\lambda,\,q\in Q_\lambda,\,pq\in C}\operatorname{sgn}(q)$$
for the irreducible characters of the ...

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### What is the effect of tensoring with the sign representation on irreducible modules for a Type D Weyl group?

Given an integer $n \geq 4$, consider the Weyl groups $W(B_n)$ and $W(D_n)$ of types $B_n$ and $D_n$, respectively, and consider their representation theory over the field of complex numbers.
The Weyl ...

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### When are these irreducible complex representations for the Type D Weyl group self-dual?

Given an integer $n \geq 4$, consider the Weyl groups $W(B_n)$ and $W(D_n)$ of types $B_n$ and $D_n$, respectively, and consider their representation theory over the field of complex numbers.
The Weyl ...

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### On characters of the symmetric group: Part 2

This question is related to my earlier MO quest.
For an integer partition $\lambda$, denote $\ell(\lambda)=$ length, $\vert\lambda\vert=$ size and $\lambda=$ conjugate of $\lambda$. Allow to write $\...

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### On characters of the symmetric group: Part 1

Given an integer partition $\lambda$, denote $\ell(\lambda)=$ length, $\vert\lambda\vert=$ size and $\lambda=$ conjugate of $\lambda$. Allow to write $\lambda\vdash n$ either as $(\lambda_1,\dots,\...

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### Interpretation of "1089-number trick" in terms of symmetric group action on cohomology group?

I tried posting the following on math.stackexchange, but no answers. I can of course delete if inappropriate.
The "1089 number trick" (see e.g. here) says that if you take a three-digit ...

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### Major indices of standard tableaux of shapes obtained from addable cells of a given Young diagram

I have a "very" indirect proof that the following fact is true for every Young diagram $\lambda \vdash n$ and every $r \in \{0,\dotsc,n\}$:
\begin{equation}
d_\lambda = \sum_{a \in \mathrm{...

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### What is $\dim D^{\lambda}$ for the symmetric group?

What are the dimensions of the simple modules $D^{\lambda}=S^{\lambda}/S^{\lambda}\cap (S^{\lambda})^{\perp}$ for the modular representation theory of $S_n$, i.e. $\operatorname{char}(k)=p>0$?
I ...

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### Asymptotics for number of $p$-regular partitions of $n$

The number of simple modules $D^{\lambda}=S^{\lambda}/S^{\lambda}\cap (S^{\lambda})^{\bot}$ of the symmetric group over a field $k$ such that $\text{char}(k)=p > 0$ is the number of $p$-regular ...

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### What is known about representations of $S_n$ in other categories?

Is anything known about representations of the symmetric group $S_n$ for categories other than $\textbf{Vect}_k$, vector spaces and linear maps over a field $k$.
That is, a group $G$ can be considered ...

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### Equivalence of dihedral and symmetric group actions on a specialized real algebra

Edit: fixed misaligned indentation for "Update x and y by", below. I also had two little ideas that might help.
consider first the case where the digit 7 is not allowed, simplifying the ...

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### Minimal dominant permutation in weak order

Consider $S_\infty$ as a Coxeter group with Coxeter generators the adjacent transpositions $s_i$, $i\geq 1$. We view elements of $S_\infty$ as functions $u:\mathbb{N}\to\mathbb{N}$. Recall the Lehmer ...

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### A question regarding symmetrizing the tensor product of vectors in two different ways

Let $V = \mathbb{C}^m$, endowed with the standard hermitian inner product which we will denote by $\langle \cdot, \cdot \rangle$, $n$ be a positive integer and $\Sigma_n$ denote the symmetric group on ...

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### Is the appearance of Schur functions a coincidence?

The Schur functions are symmetric functions which appear in several different contexts:
The characters of the irreducible representations for the symmetric group (under the characteristic isometry).
...

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### Subgroups of the symmetric group and binary relations

Motivation
The following came up in my work recently. (NB this is the motivation, not the question I'm asking. You can skip to the actual question below, which is self-contained, but not self-...

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### Basis parametrized by the symmetric group elements for the coinvariant algebra

Let $A_n$ be the coinvariant algebra of the symmetric group $S_n$.
This algebra has vector space dimension $n!$.
$A_n$ is the quotient algebra of the polynomial ring $K[x_1,...,x_n]$ by the elementary ...

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### The quotient of a higher Specht polynomial over the corresponding regular Specht polynomial

I'll need some notation before I can phrase my question, so please bare with me for a little. I'll try to get there as fast as possible (it's also my first MO question...).
Let $\lambda$ be a ...

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### Maximal subgroup in $S_{10}$

Consider the set of unordered pairs $\{(i,j)\}$, $i<j, i=1,2, \ldots, 2k+1$, $j=i+1, \dots, 2k+2$, and the group $G=S_{k(2k+1)}$ of all permutations of those pairs.
Is the subgroup of the ...

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### geometric meaning to pairs of SYT indexing for the basis of cohomology ring of full flag variety

For Grassmannians, the Schubert cells can be indexed by certain Young Tableaux, whose partition determines the dimensions of intersections of the chosen subspace with the standard complete flag. For ...

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### Decomposition of tensors into symmetry classes according to Schur functors

I am mainly looking for references on this subject, as I was unable to find any, at least any that answers what I am looking for to a satisfactory degree.
As it is well-known and extremely easy to ...

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### Twisted permutations

We consider a set $E$ with an involution (having perhaps fixed points).
We denote orbits by $\lbrace x,\overline{x}\rbrace$ (with $\overline{x}=x$ in
the case of a fixed point).
We consider sequences $...

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### The canonical automorphism of the symmetric group

Let $S_n$ be the symmetric group of order $n$. Denoting simple transpositions by $\sigma_i$ the collection $\sigma_1, \dots, \sigma_{n-1}$ generates $S_n$ subject to the following relations:
$$
\sigma ...

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### Analog of self-conjugate representation of symmetric group for Hecke algebra

Consider a symmetric group $S_n$. It is generated by generators $\sigma_1\dotsc\sigma_{n-1}$ that satisfy the following relations:
Square relations: $\sigma_k^2=1,\qquad k=1\ldots n-1$.
Braid ...

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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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### 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 matrix units 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_{...