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.
462 questions
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Is there a more natural way to define the Young symmetrizer and the Specht module?
It's well known that Young symmetrizer is a fundamental tool in the representation theory of symmetric groups.
For instance, for every Young diagram $\lambda\vdash n$, we construct a Young tableau $T_\...
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induced module of hyperoctahedral group
Let $H$ be the subgroup of the symmetric group $\mathfrak{S}_n$. Let $W_n$ be the group algebra of the hyperoctahedral group $\mathbb{Z}/2\mathbb{Z} \wr \mathfrak{S}_n$.The induced module $M:=\mathrm{...
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Where to begin in Computational Group Theory?
I'm coding a small application that looks for periodic solutions to the gravitational n-body problem. I'm trying to better understanding the symmetries of solutions, which is made up of the product of ...
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Class multiplication coefficients of symmetric groups
My question is that I was working with some counting problems, and finally the answer should be
$$
\nu_{\mu_1,\mu_2,\mu_3}=\#\{(\sigma_1,\sigma_2,\sigma_3): \sigma_1\sigma_2\sigma_3=1, \sigma_1\in C_{\...
2
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2
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Invertibility of one matrix constructed by order n subgroup of symmetric group
Let $S_n$ be the symmetric group on $n$ elements $\{ 1,2,\dotsc,n \}$ and $G$ be a subgroup of $S_n$ of order $n$. Denote the elements in $G$ by $\{ \sigma_1,\dotsc,\sigma_n \}$. Let the matrix $A=(\...
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Basis of Specht module of symmetric groups
I am reading the construction of the Specht module from James's book. The Specht module of a symmetric group corresponding to a partition $\lambda$ is spanned by all polytabloids $e_{t}$ associated ...
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Unitary representations of the symmetric group over finite fields
I am interested in understanding the unitary representations of the symmetric group over $\mathbb{F}_{q^2}$. In general, some comments here are relevant
Unitary representations of finite groups over ...
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Symmetric groups acting on rational surfaces
Let $X$ be a complex projective rational surface. Is there an upper bound on $n\in\mathbb{N}$ such that $S_n\subset \text{Aut}(X)$? Here $S_n$ is the symmetric group on $n$ elements.
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Morita equivalence between category of modules of hyperoctahedral group with the category of modules of direct product of two symmetric groups
I am reading the paper "R. Dipper and G. D. James, Representations of Hecke algebras of type $B_n$, J. Algebra (146) 1992, 454–481".
Theorem 4.18 says that the category of modules of the ...
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1
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Drunken X-mas polynomials for graphs
Given a finite connected graph $\Gamma$ with vertices $\lbrace 1,\ldots,N\rbrace$,
we can consider the polynomial
$$\sum_{\pi\in\mathcal S_N}x^{\sum_{j=1}^Nd_\Gamma(j,\pi(j))}$$
where $\mathcal S_N$ ...
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Second homotopy group of the symmetric power of a space
Let $X$ be a finite CW complex, $n \ge 2$, and $\Sigma_n$ be the permutation group on $n$ symbols. Let $X^{(n)}=X^n/\Sigma_n$ be the quotient of the natural action of $\Sigma_n$ on $X^n$. We call $X^{(...
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Representation of equivariant maps
Let $n,m,k$ be positive integers. Consider the action of symmetric group $S^n$ on $\mathbb{R}^{n\times i}$ (for $i\in \{m,k\}$) by permuting rows; i.e. for each $\pi\in S^n$ and every $n\times i$ ...
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Describing the hook part of the symmetric group algebra
Let $\mathbf{k}$ be a field of characteristic $0$. Let $n\in\mathbb{N}$, and
consider the symmetric group $S_{n}$ consisting of all permutations of
$\left[ n\right] :=\left\{ 1,2,\ldots,n\right\} $...
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Is the small Davenport constant for $S_n$, $d(S_n)=n(n-1)/2$?
The Davenport constant $D(G)$ of a finite group $G$ is the minimal $d$ such that any sequence/multiset of length $d$ is one-product, i.e., identity can be obtained as a product (in some order) of some ...
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One-product free sequences for $A_n$
I am working on computing the Davenport constant $D(G)$ for $S_n$ and $A_n$, i.e., the minimal number $d$ such that every sequence (multiset) of $d$ elements contains some subsequence giving identity ...
3
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1
answer
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Davenport constant $D(S_5)=10$ or $11$?
I am working on computing the Davenport constant $D(G)$
of symmetric groups, which is the minimal number $d$
such that every sequence of $d$
elements, possibly with repetitions, is one-product, i.e. ...
3
votes
0
answers
155
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Correspondence between even and odd permutations in $S_5$
I am working on the Davenport constant for symmetric groups, $D(G)$
, which is the minimal number $d$
such that every sequence of $d$
elements in the group G
is one-product sequence, i.e, we can ...
4
votes
1
answer
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Commutativity of the wreath product
(Cross-posted from MSE, with isomorphism replaced by conjugate: https://math.stackexchange.com/questions/4928697/commutativity-of-the-wreath-product?noredirect=1#comment10531931_4928697 )
Let $G$ be a ...
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Schur cover of alternating groups
Wilson's book "The finite simple groups" gives (in section 2.7) a description of the double cover of the alternating groups. First, one constructs a double cover $2S_n$ of the symmetric ...
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Young symmetrizers-like projections to the center of group algebra
Let $A:=\mathbb{C}S_n$ be the symmetric group aglebra.
Let $T$ be a standard Young tableaux of shape $\lambda$. Denote $R(T)$ and $C(T)$ as row and column stabilizers of $T$. For a set $S \subseteq ...
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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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1
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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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12
votes
2
answers
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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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answers
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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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votes
1
answer
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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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answers
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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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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 ...
- 2,160
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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 ...
- 2,160
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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 ...
- 301
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0
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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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answers
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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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1
answer
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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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1
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203
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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 ...
- 151
1
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1
answer
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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 ...
- 2,168
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1
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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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12
votes
1
answer
642
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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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5
votes
0
answers
200
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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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2
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376
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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 ...
- 26.5k
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0
answers
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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 ...
- 131
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1
answer
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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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