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.

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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}\...
Daniel Litt's user avatar
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72 votes
4 answers
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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 $2$. ...
Benoit Jubin's user avatar
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51 votes
3 answers
7k views

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 ...
Chris Bowman's user avatar
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48 votes
4 answers
5k views

How to constructively/combinatorially prove Schur-Weyl duality?

How is Schur-Weyl duality (specifically, the fact that the actions of the group ring $\mathbb{K}\left[ S_{n}\right] $ and the monoid ring $\mathbb{K}\left[ \left( \operatorname*{End}V,\cdot\...
darij grinberg's user avatar
36 votes
4 answers
3k views

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: $\mathrm{per}(A) = \sum_{\sigma \in S_n}\prod_{i}a_{i, \sigma(i)}$ ...
Marcin Kotowski's user avatar
30 votes
2 answers
1k views

Does the symmetric group $S_{10}$ factor as a knit product of symmetric subgroups $S_6$ and $S_7$?

By knit product (alias: Zappa-Szép product), I mean a product $AB$ of subgroups for which $A\cap B=1$. In particular, note that neither subgroup is required to be normal, thus making this a ...
John McVey's user avatar
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30 votes
0 answers
790 views

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 ...
mnmse475's user avatar
  • 301
29 votes
3 answers
4k views

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 $...
Fedor Petrov's user avatar
26 votes
6 answers
2k views

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 ...
darij grinberg's user avatar
25 votes
3 answers
3k views

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 "...
Igor Rivin's user avatar
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24 votes
5 answers
3k views

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 ...
Igor Makhlin's user avatar
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24 votes
0 answers
868 views

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 the Nekrasov-...
Dianbin Bao's user avatar
23 votes
1 answer
1k views

On an asymptotic formula of Keating and Snaith involving the Riemann zeta function

Keating and Snaith have a famous conjecture on the asymptotics of the integral $\int_0^T |\zeta(\frac 12+it)|^{2k}\, dt$, where $\zeta$ denotes the Riemann zeta function. See page 510 of the book ...
Richard Stanley's user avatar
22 votes
6 answers
2k views

What is the standard 2-generating set of the symmetric group good for?

I apologize for this question which is obviously not research-level. I've been teaching to master students the standard generating sets of the symmetric and alternating groups and I wasn't able to ...
Matthieu Romagny's user avatar
22 votes
2 answers
1k views

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: $$ ...
John Wiltshire-Gordon's user avatar
22 votes
1 answer
575 views

A symmetric-like group and the quaternion group $Q_8$

It is well known that the symmetric group $S_n$ admits presentation with $\{(ij) \mid i\neq j\}$ as the set of generators and the following list of relations (in every formula distinct letters denote ...
Sergey Sinchuk's user avatar
21 votes
2 answers
2k views

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 $i\...
darij grinberg's user avatar
21 votes
4 answers
2k views

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 ...
21 votes
1 answer
1k views

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 ...
L. T. P. L.'s user avatar
21 votes
1 answer
2k views

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, ...
darij grinberg's user avatar
21 votes
1 answer
724 views

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 $w:\mathbb{N}\rightarrow\mathbb R^+$, ...
ARupinski's user avatar
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20 votes
4 answers
2k views

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 ...
Igor Makhlin's user avatar
  • 3,493
19 votes
8 answers
14k views

"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 ...
Roland Bacher's user avatar
19 votes
1 answer
726 views

Young's natural representation of the symmetric group

The literature on the representation theory of the symmetric group contains some terminology that I find puzzling, and I am wondering if someone here knows the full story. One of the standard ways to ...
Timothy Chow's user avatar
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19 votes
0 answers
542 views

Large values of characters of the symmetric group

For $g$ an element of a group and $\chi$ an irreducible character, there are two easy bounds for the character value $\chi(g)$: First, the bound $|\chi(g)|\leq \chi(1)$ by the dimension of the ...
Will Sawin's user avatar
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18 votes
3 answers
2k views

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), ...
thibo's user avatar
  • 333
17 votes
4 answers
1k views

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 ...
John Jiang's user avatar
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17 votes
0 answers
343 views

Row of the character table of symmetric group with most negative entries

The row of the character table of $S_n$ corresponding to the trivial representation has all entries positive, and by orthogonality clearly it is the only one like this. Is it true that for $n\gg 0$, ...
Sam Hopkins's user avatar
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16 votes
2 answers
792 views

Decomposing $(\mathbb C^n)^{\otimes m}$ as a representation of $S_n\times S_m$

$V=\mathbb C^n$ is a $\mathbb CS_n$-module, where $S_n$ is the symmetric group of degree $n$, via the representation sending a permutation to the corresponding permutation matrix. The tensor power $V^...
Benjamin Steinberg's user avatar
16 votes
2 answers
703 views

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 ...
Igor Rivin's user avatar
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16 votes
4 answers
4k views

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 ...
Andy Putman's user avatar
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16 votes
1 answer
851 views

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. ...
Sean Eberhard's user avatar
16 votes
1 answer
472 views

Irreducible representations occuring in $\mathrm{Ind}_G^{S_{|G|}}1$ for $G$ finite group

Let $G$ be a finite group with $|G|=n$, let $S_G=S_n$ be the group of $n!$ permutations of the set $G$. Then $G$ is a subgroup of $S_G$ via left-translation (i.e. $g\in G$ corresponds to the ...
JoS's user avatar
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15 votes
5 answers
4k views

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 $\{1,...
15 votes
2 answers
1k views

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 ...
Steven Sam's user avatar
15 votes
1 answer
627 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 ...
Nicholas Proudfoot's user avatar
15 votes
1 answer
702 views

Schur-Weyl duality and q-symmetric functions

Disclaimer: I'm far from an expert on any of the topics of this question. I apologize in advance for any horrible mistakes and/or inaccuracies I have made and I hope that the spirit of the question ...
Saal Hardali's user avatar
  • 7,549
15 votes
2 answers
817 views

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 $(...
Nicholas Proudfoot's user avatar
15 votes
1 answer
568 views

What is the centralizer of a Young subgroup of $S_n$?

In their celebrated paper "A new approach to the representation theory of the symmetric group. II", Okounkov and Vershik prove that $Z(n-1,1)$, the centralizer of $\mathbb{C}[S_{n-1}]$ in $\...
Alvaro Martinez's user avatar
15 votes
2 answers
708 views

Gelfand-Tsetlin algebras and "Jucys-Murphy elements" for $\mathfrak{gl}_n$

I'm trying to figure out/find in literature the details concerning Gelfand-Tsetlin algebras for $\mathfrak{gl}_n(\mathbb C)$ (Okounkov-Vershik style, if you wish). Consider the chain $$\mathcal U(\...
Igor Makhlin's user avatar
  • 3,493
15 votes
1 answer
502 views

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}(\...
user148212's user avatar
  • 1,534
15 votes
1 answer
607 views

Characteristic classes of symmetric group $S_4$

For the symmetric group $S_3$, it is classically known that \begin{equation} H^*(S_3;\mathbb{Z})\cong \mathbb{Z}[x,y]/(2x,6y,x^2-3y), \end{equation} where $|x|=2$ and $|y|=4$. Moreover, $x$ can be ...
Bob's user avatar
  • 439
15 votes
1 answer
677 views

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 ...
Steven Sam's user avatar
15 votes
1 answer
711 views

Character theoretic proof of the Littlewood–Richardson rule?

The Littlewood–Richardson coefficients are the multiplicities $$ c(\lambda,\mu,\nu)= \dim_{\mathbb{C}}\operatorname{Hom}_{S_n}(S(\nu),S(\lambda/\mu)) $$ and the Littlewood–Richardson rule says that ...
Chris Bowman's user avatar
  • 1,191
14 votes
4 answers
1k views

A finite dimensional algebra associated to the symmetric group

Let $S_n$ be the finite group given as $n \times n$ permutation matrices. Define for a given field $K$ the algebra $B_n$ as the subalgebra of $M_n(K)$ generated by all permutation matrices of $S_n$. (...
Mare's user avatar
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14 votes
4 answers
2k views

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 ...
user2052's user avatar
  • 1,401
14 votes
5 answers
2k views

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 ...
Ngoc Mai Tran's user avatar
14 votes
4 answers
775 views

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 ...
rishig's user avatar
  • 143
14 votes
1 answer
897 views

Church-Farb on the cohomology of pure braid groups and character polynomials, intuition behind proof of result?

Fix $n \ge 2$. Let $V_n$ be the $\binom{n}{2}$-dimensional vector space (over $\mathbb{C}$) generated by a set of vectors $\{w_{ij} : 1 \le i < j \le n\}$. Let $\bigwedge^* V_n$ be the exterior ...
user102036's user avatar
14 votes
2 answers
1k views

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 $$(g_1,\...
Boaz Tsaban's user avatar
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