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.
36 questions from the last 365 days
6
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0
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130
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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_\...
1
vote
0
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36
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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{...
- 179
4
votes
1
answer
378
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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 ...
- 297
0
votes
0
answers
95
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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
votes
2
answers
132
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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=(\...
- 21
3
votes
0
answers
169
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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 ...
- 179
1
vote
0
answers
85
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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 ...
- 571
5
votes
0
answers
145
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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.
- 193
2
votes
0
answers
96
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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 ...
- 179
3
votes
1
answer
149
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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$ ...
- 17.9k
6
votes
0
answers
136
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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^{(...
- 311
1
vote
1
answer
95
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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$ ...
- 5,405
8
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1
answer
356
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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\} $...
- 33.8k
2
votes
1
answer
232
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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 ...
0
votes
0
answers
164
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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
votes
1
answer
260
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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
views
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
593
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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 ...
- 891
3
votes
1
answer
182
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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 ...
- 7,527
3
votes
0
answers
89
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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 ...
- 31
9
votes
2
answers
245
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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$ ...
- 1,104
0
votes
1
answer
205
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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 ...
- 111
12
votes
2
answers
882
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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 ...
- 980
3
votes
0
answers
72
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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 ...
- 311
6
votes
1
answer
228
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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 ...
- 26.5k
6
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3
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434
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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)$ ...
- 1,066
16
votes
3
answers
1k
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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 ...
- 26.5k
1
vote
0
answers
151
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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\...
- 14k
1
vote
1
answer
82
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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 ...
6
votes
0
answers
102
views
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 ...
- 2,168
3
votes
0
answers
129
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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 ...
- 891
2
votes
0
answers
213
views
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 ...
- 7,746
1
vote
0
answers
130
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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) ...
- 25
9
votes
0
answers
254
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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 ...
- 2,306
5
votes
1
answer
212
views
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
2
votes
1
answer
145
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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