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.
438
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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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A criterion for when a symmetrized decomposable tensor is nonzero
In the problem I am interested in, one has a collection of vector spaces, all $2$-dimensional, denoted by $V_{i, j}$, where $1 \leq i \neq j \leq n$. In other words, the set of indices is
$$S = \{(i, ...
1
vote
0
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Efficient decomposition algorithm for characters of symmetric groups
Let $\chi$ be a rational character of $G:=S_n$, and we want to know whether it decomposes into irreducibles $\chi_\lambda$, for $\lambda\in\Lambda$, with $\Lambda$ given, as
$$
\chi=\sum_{\lambda\in\...
1
vote
1
answer
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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
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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 ...
26
votes
6
answers
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Why is the right permutohedron order (aka weak order) on $S_n$ a lattice?
This is one of those things I never expected to be hard until I tried to prove it. Why is the right permutohedron order (a.k.a. weak Bruhat order, a.k.a. weak order -- not to be confused with the ...
3
votes
0
answers
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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 ...
4
votes
0
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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 ...
2
votes
0
answers
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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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1
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Branching rule of $S_n$ and Springer theory
Let $u\in\mathrm{GL}_n$ be a unipotent element, let $\mathcal{B}_u$ be the variety of Borel subgroups containing $u$, and let $d=\dim \mathcal{B}_u$. Then Springer theory tells us that $H^{2d}(\...
1
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0
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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) ...
8
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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 ...
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 ...
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 ...
1
vote
1
answer
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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 ...
0
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0
answers
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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 $\...
2
votes
0
answers
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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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0
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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 ...
11
votes
1
answer
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Dual of a Specht module
For a partition $\mu$ of $n$, let $S^{\mu}$ be the associated Specht module, defined over $\mathbb{Z}$. For any field $k$, we can tensor $S^{\mu}$ with $k$ to get a representation $S^{\mu}_k$ of the ...
4
votes
0
answers
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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{...
3
votes
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 ...
4
votes
0
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 ...
12
votes
7
answers
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Easy proof of the uncountability of bijections on natural numbers
Is there an easy proof of the uncountability of bijections on natural numbers?
The proof that I have in mind is as follows -
$\text{Gal }(\overline{\mathbb Q}/\mathbb Q)$ is a proper uncountable ...
3
votes
0
answers
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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 ...
25
votes
3
answers
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Simplicity of alternating group $A_n$
I am teaching an introductory group theory course, and it has come to the inevitable proof that $A_n$ is simple for $n\geq 5$. Now, there seem to be a number of proofs that I can find – one the "...
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 ...
1
vote
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 ...
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0
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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 ...
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).
...
7
votes
1
answer
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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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votes
0
answers
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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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votes
1
answer
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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. ...
7
votes
2
answers
330
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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 ...
7
votes
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 ...
5
votes
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 ...
5
votes
1
answer
110
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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 ...
9
votes
1
answer
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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 ...
7
votes
2
answers
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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 ...
3
votes
0
answers
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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 $...
2
votes
0
answers
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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 ...
12
votes
1
answer
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Question on a reduction in Kirillov's paper on positivity of divided difference operators
As the title says, my question is on a specific argument in Kirillov - Skew divided difference operators and Schubert polynomials (journal, MSN) on positivity of divided difference operators. I recall ...
6
votes
1
answer
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Can Matsumoto's theorem for the symmetric group be proved using a monovariant?
This is a question that can be asked for any Coxeter group, but for the sake of simplicity I will restrict myself to symmetric groups. Recall the main definitions:
Let $n$ be a nonnegative integer. ...
0
votes
1
answer
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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 ...
2
votes
2
answers
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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 ...
7
votes
1
answer
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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:\...
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 ...
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votes
8
answers
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"Natural" generating sets for symmetric groups
The symmetric group on $n$ letters has
many sets of generators. Some of them are more natural than others, eg the
set $(i,i+1)$ of adjacent transpositions (natural with respect to the type A Weyl ...
0
votes
0
answers
105
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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(...
0
votes
1
answer
617
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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] &= (...
8
votes
2
answers
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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 ...