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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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 ...
alpha2357alpha's user avatar
6 votes
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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 ...
Matt Samuel's user avatar
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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 ...
eti902's user avatar
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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 ...
Asvin's user avatar
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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) ...
Yuting Li's user avatar
8 votes
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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 ...
Hjalmar Rosengren's user avatar
5 votes
1 answer
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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 ...
Christopher Drupieski's user avatar
2 votes
1 answer
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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 ...
Christopher Drupieski's user avatar
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201 views

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 $\...
T. Amdeberhan's user avatar
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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,\...
T. Amdeberhan's user avatar
29 votes
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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 ...
mnmse475's user avatar
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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 ...
Jackson Walters's user avatar
3 votes
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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 ...
Jackson Walters's user avatar
3 votes
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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 ...
Jackson Walters's user avatar
1 vote
1 answer
119 views

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 ...
Dement's user avatar
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1 vote
1 answer
106 views

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 ...
Matt Samuel's user avatar
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8 votes
1 answer
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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 ...
Malkoun's user avatar
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11 votes
1 answer
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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). ...
matha's user avatar
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5 votes
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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-...
Z. A. K.'s user avatar
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7 votes
2 answers
329 views

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 ...
Mare's user avatar
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7 votes
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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 ...
Shaul Zemel's user avatar
5 votes
1 answer
234 views

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 ...
Анатолий Вершик's user avatar
5 votes
1 answer
110 views

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 ...
staedtlerr's user avatar
7 votes
2 answers
256 views

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 ...
Bence Racskó's user avatar
3 votes
0 answers
106 views

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 $...
Roland Bacher's user avatar
2 votes
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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 ...
Jake Wetlock's user avatar
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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 ...
V. Asnin's user avatar
2 votes
2 answers
206 views

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 ...
user488802's user avatar
7 votes
1 answer
318 views

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:\...
Q. Zhang's user avatar
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1 vote
1 answer
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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 ...
Roland Bacher's user avatar
7 votes
0 answers
97 views

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 ...
darij grinberg's user avatar
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0 answers
104 views

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(...
Sergii Voloshyn's user avatar
0 votes
1 answer
609 views

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] &= (...
Dan1618's user avatar
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3 votes
1 answer
282 views

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 ...
tomatosoup's user avatar
3 votes
0 answers
49 views

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 ...
Malkoun's user avatar
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4 votes
1 answer
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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 $...
constantine's user avatar
7 votes
1 answer
309 views

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. ...
dmitry's user avatar
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19 votes
0 answers
535 views

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 ...
Will Sawin's user avatar
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9 votes
1 answer
824 views

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 ...
dm82424's user avatar
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8 votes
0 answers
211 views

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 ...
babu_babu's user avatar
  • 229
7 votes
0 answers
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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 ...
richrow's user avatar
  • 357
2 votes
0 answers
190 views

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_{...
dm82424's user avatar
  • 350
9 votes
2 answers
775 views

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 ...
Trevor K's user avatar
1 vote
1 answer
179 views

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 ...
Yachen Liu's user avatar
3 votes
0 answers
141 views

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: $$ X_i = \displaystyle \sum_{k=1}^{i-1} (k,i) $$ ...
eti902's user avatar
  • 835
2 votes
0 answers
130 views

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 ...
V. Asnin's user avatar
4 votes
0 answers
207 views

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 ...
谁家的鸡's user avatar
8 votes
2 answers
261 views

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 ...
Didier de Montblazon's user avatar
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 ...
Jingjie Yang's user avatar

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