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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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: $\Omega^\infty S^\infty:=\varinjlim~ \Omega^nS^n$ ...
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Is ${\rm S}_6$ the automorphism group of a group?

The automorphism group of the symmetric group $S_n$ is $S_n$ when $n$ is not $2$ or $6$, in which cases it is respectively $1$ and the semidirect product of $S_6$ with the (cyclic) group of order $2$. ...
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What to do now that Lusztig's and James' conjectures have been shown to be false?

Lusztig and James provided conjectures for dimensions of simple modules (or decomposition numbers) for algebraic groups and symmetric groups in characteristic $p$. These conjectures have been ...
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What are the applications of immanants?

Definitions of determinant: $\det(A) = \sum_{\sigma \in S_n}(-1)^{\operatorname{sgn} \sigma}\prod_{i}a_{i, \sigma(i)}$ and permanent: $\mathrm{per}(A) = \sum_{\sigma \in S_n}\prod_{i}a_{i, ...
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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 $w:\mathbb{N}\rightarrow\mathbb R^+$, ...
19
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Conjectural identities for Young symmetrizers and Young-Jucys-Murphy elements

The following questions I have found in my own notes from about 3 years ago. Unfortunately, I lost much of the context; I believe I made these conjectures reading Okounkov-Vershik, arXiv:0503040v3, ...
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Nekrasov-Okounkov hook length formula

I am now reading the paper An explicit expansion formula for the powers of the Euler Product in terms of partition hook lengths by Guo-Niu Han. The author rediscovered what he calls Nekrasov-Okounkov ...
18
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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 ...
18
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An n!xn! 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 ...
18
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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 ...
18
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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: $$ ...
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What is the length of the shortest law of $S_n$?

What is the length of the shortest word $w\in F_2$ such that $w(x,y)$ is trivial for every $x,y\in S_n$? There is a simple argument showing that we must have $\ell(w)\geq n$. See here for instance. ...
16
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1answer
255 views

Cohomology of configuration space as a representation of the symmetric group

Let $X_n$ be the space of $n$ distinct labeled points in $\mathbb{R}^3$, which is equipped with an action of the symmetric group $S_n$. It is well known that the total cohomology of $X_n$ is ...
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Has Reifegerste's Theorem on RSK and Knuth relations received a slick proof by now?

For the notations I am using, I refer to the Appendix at the end of this post. Here is what, for the sake of this post, I consider to be Reifegerste's theorem: Theorem 1. Let $n\in\mathbb N$ and ...
16
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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 ...
16
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Okounkov-Vershik approach to representation theory of $S_n$

This is a rather soft question. I was wondering if someone could explain on a fundamental and intuitive level, what the Okounkov-Vershik approach to representation theory of $S_n$ is all about. It's ...
15
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Minimal maximal subgroup of the symmetric group

The question is pretty much in the title: What is the maximal subgroup of $S_d$ of maximal index (so minimal size)? A slight variant (I am not sure if it leads to a different answer) is: what if we ...
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roots of permutations

Consider equation $x^2=x_0$ in symmetric group $S_n$, where $x_0\in S_n$ is fixed. Is it true that for each integer $n\geq 1$, the maximal number of solutions (square roots) has identity permutation? ...
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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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Largest permutation group without 2-cycles or 3-cycles

The largest permutation group without 2-cycles is $A_n$, which has size $n!/2$. I think the largest permutation group without 2-cycles or 3-cycles is much smaller, but I can't figure out if it should ...
14
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factorization of the regular representation of the symmetric group

Let $\mathbb{C}[S_n]$ be the regular representation of the symmetric group $S_n$, and let $\mathbb{C}^n$ be the vector representation. Question: Does there exist a representation $V$ (of dimension ...
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Number of squares in a finite group

This was asked at MSE but never answered. Let $G$ be a finite group and denote by $sq(G)$ the number of squares in $G$ i.e. the number of elements in $G$ which possess a square root. For example, if ...
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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 ...
13
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Where do stable Kronecker coefficients live “in nature”?

Background: For a partition $\lambda$, let $\lambda[N] = (N - |\lambda|, \lambda_1, \lambda_2, \lambda_3, \dots)$, also let $\chi_\lambda$ be the corresponding irreducible character of the symmetric ...
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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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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, ...
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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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Connected components $0-1$ matrices

Let $M$ be a $0-1$ matrix. Here a matrix has one component means we can traverse from a matrix entry $(i,j)$ which is $1$ to any other one by moving step of $(i\pm1,j),(i,j\pm1),(i\pm1,j\pm1)$ where ...
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The Simultaneous Conjugacy Problem in the symmetric group $S_N$

We are interested in the following notions in the case $G=S_N$, the symmetric group on $\{1,\dots,N\}$. Fix a group $G$ and a number $d$. For $(g_1,\dots,g_d)\in G^d$ and $x\in G$, define ...
11
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Can a large transitive permutation group need many generators?

let $G$ be a transitive permutation group acting on $\{1, \ldots, n\}$, and let $d(G)$ be the minimal number of generators of $G$. Is it true, that for $n\rightarrow\infty$ we have ...
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What is the probability that two random permutations have the same order?

I am interested in the orders of random permutations. Since the law of the logarithm of the order of a permutation converges to a normal law (for instance Erdös-Turan Statistical group theory III), ...
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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 ...
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What Is The Minimal Monomial of the Symmetric Group?

In the symmetric group $S_n$ what is the shortest sequence $c_1,\ldots,c_k\in S_n$ such that, for all $x\in S_n$ the following product of conjugates of $x$: $$x^{c_1}x^{c_2}\ldots x^{c_k}$$ equals the ...
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Invariant ring of $S_5$

The irreducible representations of the Symmetric group $S_5$ are classified by the partitions of $5$. For the standard representation which corresponds to the partition (4,1) the ring of invariants is ...
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Reference request: The stable Kronecker ring is isomorphic to the ring of symmetric polynomials

Background For $\lambda$ any partition and $n$ a positive integer, write $\lambda[n]$ for the sequence $(n - |\lambda|, \lambda_1, \lambda_2, \ldots, \lambda_r)$. For $n$ large enough, this is a ...
10
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A $q$-analogue of Foulkes' character related to alternating permutations

My paper "Alternating permutations and symmetric functions" at http://math.mit.edu/~rstan/papers/altenum.pdf enumerates certain classes of alternating permutations, such as those whose inverse is ...
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Number of Permutations?

Edit: This is a modest rephrasing of the question as originally stated below the fold: for $n \geq 3$, let $\sigma \in S_n$ be a fixed-point-free permutation. How many fixed-point-free permutations ...
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How do most people write permutations?

I'd like to know how people prefer to write permutations, or elements of the symmetric group $S_n$ for $n\ge0$. The most natural way to define a permutation in $S_n$ is as a bijection on the set ...
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Problems which use S₄ → S₃

I need examples of problems which use, directly or indirectly, the homomorphism $S_4\to S_3$ in the solution (its kernel is $\mathbb{Z}_2\oplus\mathbb{Z}_2$). Obvious candidates: Lagrange resolvent ...
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Characters of permutation groups

Let $N$ be a fixed positive integer, and denote by $C(m)$ the number of permutations on an $N$-element set that have exactly $m$ cycles (counting $1$-cycles). Then it is in the literature that the ...
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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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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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On Applications of Murnaghan Nakayama Rule

This question is crossposted at math.stackexchange here and may be beyond the usual scope of the site. The question is located below. In short, I am looking for an accessible explanation of the ...
8
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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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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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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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Is there a non-explicit characterization of the Specht modules?

It is a basic fact about the symmetric group $S_n$ that its irreducible representations are indexed by partitions of $n$. My question is, can the association between partitions and irreps be ...
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Linear relations among permutation matrices

Given a permutation $\sigma\in S_n$, let $P_\sigma$ denote the corresponding $n\times n$ permutation matrix. It is easy to see that for $n=3$, there is only one linear relation up to scaling given by ...
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What is natural about the well-known bijection between conjugacy classes and irreps of a symmetric group?

Symmetric groups possess a well-known bijection between conjugacy classes and irreducible representations. More precisely, both sets are indexed by Young diagrams. Question: To what extent is this ...
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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 ...