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.

learn more… | top users | synonyms

11
votes
2answers
517 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,\...
14
votes
5answers
1k 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 ...
6
votes
4answers
901 views

Structure of the adjoint representation of a (finite) group (Hopf algebra) ?

Every group acts on itself by conjugation $h \mapsto g h g^{-1}$. Respectively considering functions on a group we obtain a linear representation. Question 1: what is known about this representation ...
2
votes
0answers
742 views

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 ...
85
votes
4answers
4k views

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}\...
38
votes
3answers
3k 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 ...
14
votes
2answers
2k views

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? ...
16
votes
2answers
1k 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\...
14
votes
2answers
487 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 $(...
18
votes
2answers
764 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: $$ ...
9
votes
2answers
361 views

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 ...
8
votes
2answers
705 views

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 ...
15
votes
1answer
276 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 ...
12
votes
4answers
1k views

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 ...
12
votes
2answers
749 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 ...
8
votes
1answer
1k views

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 ...
5
votes
1answer
642 views

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 ...
5
votes
1answer
323 views

Which subgroup order of the symmetric group is the most frequent?

Question: What is the most frequent order of subgroups of $S_n$? More precisely: Let $a_k$ be the number of subgroups of $S_n$ with order $k$. What is the maximum of $a_k$? This question came up ...
8
votes
1answer
338 views

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 ...
5
votes
2answers
355 views

A question about homogenous polynomials of degree $\frac{n(n-1)}{2}$

Let $n$ be a positive integer and $S_n$ be the symmetric group on $\{1,2,\ldots,n\}$. For any $w\in S_n$ and polynomial $f\in \mathbb{R}[x_1,x_2,\ldots,x_n]$, denote $w(f)=f(x_{w(1)},x_{w(2)},\ldots,...
1
vote
0answers
171 views

What is the definition of plethysm in the representation theory of permutation groups

Let $s_\lambda \circ s_\mu$ be a plethysm. Here let $\lambda, \mu$ be $m,n$ box Young diagrams. I have seen the definition of plethysms in symmetric functions. I would like to understand the ...