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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Distances on generalizations of the symmetric group
I'm a computer vision student, and I'm looking for some symmetric group literature guidance. I'm going to provide some context, and finally ask two questions.
The Cayley distance and other distances ...
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Necklaces and the generating function for inversions
The problem of Necklaces is well-known, i.e "The number of fixed necklaces of length $n$ composed of $a$ types of beads $N(n,a)$" can be calculated:
http://mathworld.wolfram.com/Necklace.html
Let us ...
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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 ...
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A product identity for partitions
For a partition $\lambda=(\lambda_1\ge \lambda_2\ge \dots)$, let
$m_\lambda=\prod_i (\lambda_i-\lambda_{i+1})!$ be the product of factorials of consecutive differences and let $v_\lambda=\prod_{i | \...
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Schur Weyl duality for sl_n representations
Consider a finite dimensional vector space $V$ and the general linear group $GL(V)$ acting on it. Both $GL(V)$ and the symmetric group $S_d$ act on the tensor product of $d$ copies of $V$, and by Weyl ...
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Derived functors of symmetric powers
What do the derived functors of the symmetric powers look like? I understand that this is related to the homology of the symmetric groups, but I don't know a reference for that.
Namely, I'm ...
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The Weyl group of $SL(2, F)$
Let $G= SL(2, F)$, given a torus $T$, the Weyl group with respect to $T$ is defined to be $W=N(T)/Z(T)$, the quotient of the normalizer $N(T)$ of the torus by the centralizer $Z(T)$ of the torus.
My ...
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Partitions limit shape and LDP
Hello!
I am trying to understand the paper of Dembo-Vershik-Zeitouni, Large deviations for integer partitions. I am only interested in Theorem 2, which deals with the case of the uniform distribution....
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Finding a maximal tableau for a sum of Jucys-Murphy Elements
Let $X \in \mathbb{C}[S_n]$ be an element of the group algebra of $S_n$ expressible as the sum of some Jucys-Murphy elements. Then let $\lambda$ be any irreducible representation of $S_n$, with the ...
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What are the relation between Rep(G) and Rep(S_n)?
Let G be a finite group. We know it can be written as a subgroup of S_n. On the other hand, people sometimes say Rep(G) --- the category of all finite dimensional representations, are more interesting ...
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$\lambda$-rings and hopf-rings
The direct sum of complex representation rings $R_*\oplus R\Sigma_n$, for $\Sigma_n$ the $n$th symmetric group is also the free $\lambda$-ring on one generator. Here, we take a product obtained from ...
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Partial order on partitions and symmetric group algebra
Let $n$ be a natural number. Consider a set $\Lambda_n$ of partitions of $n$ into a sum of natural numbers, like $n = \lambda_1 + \cdots +\lambda_k$ (A set of small lambdas representing a partition is ...
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Generalizing the Fundamental Theorem of Symmetric Polynomials
The fundamental theorem of symmetric polynomials tells us that the ring $\mathbb{Z}[x_1,\ldots,x_n]^{S_n}$ of symmetric polynomials in $n$ variables is generated (without relations) by the elementary ...
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Faithful transitive actions by large groups on small sets
How large is the largest transitive subgroup of $S_n$ other than itself and $A_n$? In particular, does its size grow at least exponentially in $n$?
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An $n!\times n!$ determinant
Let us consider the matrix $A$ with its rows and columns enumerated by the elements of $S_n$ with $A_{\sigma\tau}=x^{c(\sigma\tau^{-1})}$ where $c()$ is the number of cycles in a permutation's ...
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universality of Macdonald polynomials
I have been recently learning a lot about Macdonald polynomials, which have been shown to have probabilistic interpretations, more precisely the eigenfunctions of certain Markov chains on the ...
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An n!-dimensional representation of the symmetric group S_{n+2}
I have come across a sequence of representations $V_n$ of the symmetric group $S_{n+2}$ which has the property that restricting the action $S_n \subset S_{n+2}$ gives the regular representation:
$$ ...
- 5,898
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Why are Jucys-Murphy elements' eigenvalues whole numbers?
The Jucys-Murphy elements of the group algebra of a finite symmetric group (here's the definition in Wikipedia) are known to correspond to operators diagonal in the Young basis of an irreducible ...
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Formula for the "integral form" action of Iwahori-Hecke algebra on the standard basis for Specht modules
Is there a formula somewhere in the literature for the action of the generators $T_1,\ldots,T_{n-1}$ of the Iwahori-Hecke algebra on the standard basis of its Specht modules? It is well-known that the ...
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Restriction of characters of hyperoctahedral groups.
The hyperoctahedral group $H_n$ has several descriptions; as a wreath product; as signed permutation matrices; as the Weyl group of type $B_n$ or $C_n$. In all these descriptions it is apparent that ...
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Rhombus tilings with more than three directions
The point of this question is to construct a list of references on the following subject: Fix vectors $v_1$, $v_2$, ..., $v_g$ in $\mathbb{R}^2$, all lying in a half plane in that cyclic order. I am ...
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What do the stable homotopy groups of spheres say about the combinatorics of finite sets?
The Barratt-Priddy-Quillen(-Segal) theorem says that the following spaces are homotopy equivalent in an (essentially) canonical way:
$\Omega^\infty S^\infty:=\varinjlim~ \Omega^nS^n$
$\mathbb{Z}\...
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Two ways of generalizing factorials via symmetric groups
By the Bruhat decomposition of $GL(n, \mathbb{F}_q) / B_n$ we know that $$[n]! = \sum_{ \sigma \in S(n)} q^{l(\sigma)}$$ where $[n]! = \prod_{j=1}^n (1+q + \cdots + q^{j-1})$ and $l(\sigma)$ is the ...
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Is there an easy description of the structure of this infinite group?
Let $S_\infty$ denote the full symmetric group on countably many points and index the points by $\mathbb{N}$. For any weight (for lack of a better name) function $w:\mathbb{N}\rightarrow\mathbb R^+$, ...
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Tensor products of permutation representations of symmetric groups.
I am looking for a reference for the following fact which must be classical (which makes it harder, for me, to track a reference down). I am interested because there are similar (more complicated) ...
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Difference between orthogonal form and seminormal form
Frequently in the literature on Hecke algebras for the symmetric group and their generalisations, one encounters references to Young's seminormal form and Young's orthogonal form. I have a good ...
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What is known about the centraliser of the Hecke algebra in the affine Hecke algebra?
This question is a sequel to
66602
The Hecke algebra is the quotient of the group algebra of the braid group of type $A_n$ by quadratic relations and the affine Hecke algebra is the quotient of the ...
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Interpreting the Garnir relations in terms of the Yang-Baxter equation
One possible construction of the Specht modules goes as follows.
Given a partition $\lambda$ of $n$, we can write down Young's seminormal form for the representation of $S_n$ corresponding to $\...
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Picking $n$ so that certain Schur functors of the standard representation of $S_n$ are linearly independent
Let $V_n$ be the standard permutation representation of the symmetric group $S_n$, and let $\mathbb{S}_{\lambda}$ denote the Schur functor associated to the partition $\lambda$.
Let $\lambda$ range ...
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What are the applications of immanants?
Definitions of determinant:
$\det(A) = \sum_{\sigma \in S_n}\operatorname{sgn} \sigma\prod_{i}a_{i, \sigma(i)}$
and permanent:
$\mathrm{per}(A) = \sum_{\sigma \in S_n}\prod_{i}a_{i, \sigma(i)}$
...
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Irreducible decomposition of tensor product of irreducible $S_n$ representations
Are there well known results on the irreducibles in the decomposition of tensor products of irreducible $S_n$ representations? I would also like to know of some references where I can find formulas (...
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Lie algabra of symmetric group
It's easy to see that the descending central series of a group induces a graded Lie algebra .(see for example Serre's Harvard lectures or Magnus-Solitar book). I think in general this can be ...
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Cyclic Permutations - but not what you think
This question is not about elements of $S_n$ that consist of a single $n$-cycle, though naturally it's related.
Instead, consider permutations modulo the action of $(123\ldots n)$. That is, we ...
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Frobenius norms of Fourier coefficients of the symmetric group
Suppose $f \colon S_n \rightarrow \mathbb{R}$ is some weighted collection of permutations. We want to understand how "well-spread" $f$ is. Our first test is its actions on singletons - are ...
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2
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Convolution on symmetric group Sn
I have question regarding convolution of functions (say
g and h) defined on Sn. In Fourier space this is equivalent to IFT(G.H),
where G = FT(g) and H = FT(h).
Fast Fourier transforms (Clausen's FFT) ...
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sum of the character of the symmetric group
Suppose $\mu$ is a fixed partition of $n$ of length $l(\mu)$, and I was encountered with the following sum, namely
$\sum_{\nu} \chi_{\nu}(\mu)$.
I did some calculation using the character table that ...
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What is a Specht module?
I'm studying the structure of the Specht module for $S_n$ and I would like to know if there is some generalizations of this structure for Weyls groups or Coxeter groups.
Also, I'm interest to know ...
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Decomposition of induced representations in S_n
Let C be a cyclic subgroup of S_n.
How does the representation $Ind_C^{S_n}\rho$, where $\rho$ is some representation of $C$, decompose into irreducible components?
Is there are a way to know which ...
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Reference for Hecke algebra version of Young's orthogonal basis
In the paper Seminormal representations of Weyl groups and Iwahori-Hecke algebras, Arun Ram defines a seminormal basis as follows:
given a chain of split semisimple $K$-algebras $K\cong H_0 \subseteq ...
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Double coset representatives and structure of hecke algebras
Let $GL_n(F_q)$ be the general linear group over finite field $F_q$ and $B_n$ be its borel subgroup consisting of all upper triangular matrices. Then the double cosets $B_n\backslash GL_n(F_q)/B_n$ ...
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Can this nested sum be expressed in terms of generalized harmonic numbers and the cycle index polynomials of the symmetric groups?
For a paper I was working on recently I needed to find the value of the following sum:
$$S(n,k) = \sum_{i_1 = 1}^n \sum_{i_2 = i_1+1}^n \cdots \sum_{i_k=i_{k-1}+1}^n \frac{1}{i_1 i_2 \cdots i_k}.$$
...
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Branching rule from symmetric group $S_{2n}$ to hyperoctahedral group $H_n$
Embed the hyperoctahedral group $H_n$ into the symmetric group $S_{2n}$ as the centralizer of the involution $(1, 2) (3, 4) \cdots (2n-1, 2n)$ (cycle notation). Label representations of $S_{2n}$ by ...
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Roots of permutations
Consider the equation $x^2=x_0$ in the symmetric group $S_n$, where $x_0\in S_n$ is fixed. Is it true that for each integer $n\geq 0$, the maximal number of solutions (the number of square roots of $...
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A general formula for the number of conjugacy classes of $\mathbb{S}_n \times \mathbb{S}_n$ acted on by $ \mathbb{S}_n$
$\def\S{\mathbb{S}}$ Dear all,
So I have $\S_n$ acting on $\S_n \times \S_n$ via conjugacy. That is:
for $g \in \S_n, (x,y) \in \S_n \times \S_n$: $g(x,y) = (gxg^{-1},gyg^{-1}).$
Is there a general ...
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Decompositions for Symmetric Groups
I'm looking for information about how representations of $S_n$ decompose under restriction.
I know about the branching rule: That is, in characteristic 0, irreducible modules $L(\lambda)$ for $S_n$ ...
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A sum involving irreducible characters of the symmetric group
Recently, during the research, I came across a sum, denoted by $H(n,L)$, involving irreducible characters of the symmetric group,
\begin{equation}
H(n,L)\colon=\sum_{Y_{i,j,w}} \frac{\chi^{Y_{i,j,w}([...
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2
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Diagonalizing some matrices arising from Fourier transform on $S_n$.
Consider the function $f$ on $S_n$ which equals $1/n$ on all adjacent transpositions $(i,i+1)$, where we let $n+1 = 1$, and $0$ otherwise, and its Fourier transform $\hat{f}(\rho)$ evaluated at the ...
- 4,466
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1
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some confusion about the explicit construction of irreducible representations of $S_n$
In this book chapter, the irreducible representations of the symmetric group $S_n$ is given in terms of polytabloids of a Ferrer's diagram $\lambda$, defined as
$e_t = \sum_{\pi \in C_t} \text{sgn}(\...
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4
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Making the branching rule for the symmetric group concrete
This question concerns the characteristic $0$ representation theory of the symmetric group $S_n$. I'm a topologist, not a representation theorist, so I apologize if I state it in an odd way.
First, a ...
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How ugly is the isomorphism R[GxH] = R[G] (X) R[H] for groups G, H?
Clearly, when $G$ and $H$ are two finite groups, and $V$ and $W$ are two representations of $G$ and $H$, respectively, then $V\otimes W$ is a representation of the group $G\times H$. It is a well-...
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