**16**

votes

**2**answers

921 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 ...

**5**

votes

**1**answer

330 views

### 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 $n$?

**10**

votes

**0**answers

116 views

### 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 ...

**2**

votes

**0**answers

38 views

### Is $LIS(\pi)+LIS(\sigma)+LIS(\sigma\pi^{-1})$ lower bounded?

In the title, $LIS$ stands for the length of longest increasing subsequence and Greek letters stand for permutations from symmetric group $S_n$.
Considering some cases such as ...

**1**

vote

**0**answers

61 views

### research on the structure/properties of permutation matrix/table with $(i,j)th$ entry as $\pi_j\circ \pi_i^{-1}$

Is there any research on the structure/properties of permutation matrix/table with $(i,j)th$ entry as $\pi_j\circ \pi_i^{-1}$, where $\{\pi_1,\pi_2,...,\pi_{k!}\}=S_k$?
I know if we apply the ...

**16**

votes

**1**answer

652 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, ...

**2**

votes

**0**answers

101 views

### Special sets of involutions generating ${\rm S}_n$

For which positive integers $k$ and $r$ are there involutions $g_{n,i} \in {\rm S}_n$
$(n \in \mathbb{N}, \ i = 1, \dots, k)$ such that the following hold?:
for any $n$, the $g_{n,i}$ $(i = 1, ...

**11**

votes

**1**answer

452 views

### 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 ...

**2**

votes

**2**answers

318 views

### Polynomials of low degree that clone polynomials of higher degree

Let $f(x_1,\dots,x_{16})=(x_1+x_2+x_3+x_4)(x_5+x_6+x_7+x_8)(x_9+x_{10}+x_{11}+x_{12})(x_{13}+x_{14}+x_{15}+x_{16})\in\Bbb R[x]$.
Let $\mathcal{Z}$ be the zero set of $f$ in ...

**0**

votes

**0**answers

82 views

### Explicit basis/weight vectors for irreducibles inside the plethysm $Sym^m(\bigwedge^p \mathbf(V))$

This is a follow up to this question about finding the multiplicities of irreducible representations restricted to Young diagrams of 2-columns or less, inside the plethysm $Sym^m(\bigwedge^p ...

**4**

votes

**0**answers

173 views

### In general, are 'Young symmetrisers' given by Littlewood-Richardson 'Orthogonal projection Operators'?

Consider $V^{\otimes n}$ where $V$ is vector space and the representation of GL(V) acting in the usual way. Now if I consider tensor products or plethysms of irreducible spaces, this is not in general ...

**18**

votes

**6**answers

657 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 ...

**4**

votes

**0**answers

63 views

### Name for class of flattening permutations

Let $S_n$ be the symmetric group. For any sequence of numbers $y=[y_1,y_2,\cdots,y_k]$, define the flattening operation as $\mbox{flatt}_{k}(y)$ as a relabeling of $y_1,y_2,\cdots,y_k$ in terms of ...

**3**

votes

**1**answer

51 views

### Counting a Modified Class of Standard Young Tableau

Let $\lambda=(\lambda_1,\cdots,\lambda_n)$ be a partition, with $|\lambda|:=N$. Attach an extra box to $\lambda$ to the right end of the $r$'th row. In coordinate form, the last box on row $r$ has ...

**1**

vote

**1**answer

149 views

### Homomorphisms from irreducible spaces to reducible spaces

Let $P_{\lambda}$ be a Young symmetriser associated to the following tableau $(a_1 a_2 a_3 b_3 ; b_1 b_2)$ where the entries seperated by the ; belong to first and second COLUMNS of the tableau. Take ...

**4**

votes

**2**answers

164 views

### Isotypic components of the action of the symmetric group on polynomials

The polynomial ring $\mathbb{C}[x_1,\ldots,x_n]$ decomposes as a direct sum of isotypic components for the action of the symmetric group $S_n$. The isotypic component of the trivial representation is ...

**3**

votes

**1**answer

225 views

### Minimum word length for an unusual set of generators of the symmetric group

Problem. Let $n\geq 2$ and let $T$ be the set of all permutations in $S_n$ of the form
$$t_k:=\prod_{1\leq i\leq k/2}(i,k-i) \qquad \hbox{for $k=3,4,\ldots,n+1$}.$$
Find the least integer ...

**2**

votes

**1**answer

111 views

### Number of double cosets of a Young subgroup

Let $\lambda\vdash n$ be a partition of $n$ with $k$ parts and $S_\lambda$ be a Young subgroup of $S_n$.
Further let $S_\lambda\backslash S_n/ S_\lambda$ be the set of double-cosets. Now I would like ...

**5**

votes

**1**answer

175 views

### Is there a bijection of permutations onto mathematical objects that preserve information about descents?

$\omega \in S_n$ is an FPFI (fixed point free involution) (also called a matching) if $\omega^2=1$ and $\omega(i) \neq i$ for all $i$.
For $\omega \in S_n$, a descent occurs at $i$ if $\omega(i+1) ...

**2**

votes

**0**answers

634 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 ...

**2**

votes

**1**answer

146 views

### Symmetric invariants of a Schur Module

Let $V\cong\mathbb C^n$ be a complex vector space of dimension $n$. Let $\lambda\in\mathbb Z^r$ be a generalized integer partition $\lambda_1\ge\cdots\ge\lambda_r$ with $r\le n$. Denote by $\mathbb ...

**2**

votes

**1**answer

151 views

### counting the number of ordered pairs in a permutohedron

Recall that a permutohedron is a graph on the set of permutations $S_n$ with an edge between $\sigma$ and $\tau$ if they differ by one adjacent transposition: $\tau = (i,i+1) \circ \sigma$ for some $i ...

**5**

votes

**2**answers

374 views

### 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 ...

**1**

vote

**1**answer

160 views

### A Simple Bijective Proof Of Stanley's Hook-Content Formula for Hook Shapes

this is my first post on math overflow so I hope it goes well. I believe I have a fairly simple bijective proof for Stanley's Hook-Content Formula in the case of hook shapes. I wanted to see if ...

**2**

votes

**2**answers

200 views

### Minimal *-idempotents for the group algebra of the symmetric group

There is a well-known construction of minimal idempotents in the group algebra of the symmetric group $\mathbb C[S_n]$ using row symmetrizers and column antisymmetrizers. But these idempotents are ...

**0**

votes

**0**answers

94 views

### Geometric interpretation of table with permutations and inversions

Let $T(n,k)$ is the number of permutations of numbers $1, ..., n$ and each of the permutations has $k$ inversions. We can consider a table for $T(n,k)$ for some $n$ and $k$. For eg.
$n=1,...,6$, ...

**35**

votes

**3**answers

2k 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 ...

**9**

votes

**1**answer

323 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
...

**14**

votes

**5**answers

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 ...

**7**

votes

**1**answer

482 views

### Permutation character of the symmetric group on subsets of certain size

The symmetric group $S_n$ acts on $[n]:=\{1,\ldots,n\}$, thereby inducing an action on the set $$\wp_k(n)=\{\: A\subseteq[n] \::\: \#A=k \:\}$$ of subsets of cardinality $k$, simply by $$(g,A)\mapsto ...

**4**

votes

**1**answer

207 views

### Expression of basis vectors of permutation modules in different bases.

This is a cross-post from math.se, because I did not get any answer there:
Write $[n]:=\{1,\ldots,n\}$. For a partition $\lambda\vdash n$, I will write $[\lambda]$ for the Specht module that ...

**9**

votes

**4**answers

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 ...

**9**

votes

**0**answers

396 views

### 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
...

**6**

votes

**3**answers

341 views

### Representations of S_n induced from centralizers of elements

Does anyone have a reference for a good description of representations of $S_{n}$ obtained by inducing up from $C_{S_{n}}(\pi)$, for some element $\pi$ of $S_{n}$? (I'd prefer an efficient ...

**4**

votes

**4**answers

561 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

**0**answers

47 views

### What groups of symmetry are most suited for filling uniformely a spherical 3D space, whilst possessing the lowest possible surface-to-volume ratio?

I am looking for the closest known approximate solution to Kelvin foams problem that would obey a spherical symmetry.
One alternative way of formulating it: I am looking for an equivalent of ...

**11**

votes

**4**answers

1k 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, ...

**2**

votes

**2**answers

287 views

### The Jantzen-Schaper theorem

Does anybody have an electronic copy of Schaper's PhD thesis:
K.D. SCHAPER, ‘Charakterformeln fur Weyl-Moduln und Specht-Moduln in Primcharacteristik’,
Diplomarbeit, Bonn, 1981.
I would like to ...

**9**

votes

**3**answers

623 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 ...

**6**

votes

**3**answers

2k views

### Sarrus determinant rule: references, extensions

SEEKING REFERENCES FOR SARRUS' RULE AND EXTENSIONS
An undergraduate came to me with an identity for 4x4 determinants that is actually correct:
$\det(A)=h(A)+h(RA)+h(R^{2}A)$
where R cyclically ...

**5**

votes

**2**answers

246 views

### How many ways can a given permutation be obtained as a product of k 2-cycles?

Let $\sigma_1, \ldots, \sigma_b$ be all the 2-cycles in $S_n$. (So, $b = \binom{n}{2}$.) Given $\pi \in S_n$, what is known about how many ways $\pi$ can be obtained as a product of $k$ (not ...

**9**

votes

**4**answers

874 views

### 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 ...

**6**

votes

**2**answers

528 views

### Word length in the symmetric group

Let $n \geq 1$ and let $H_n$ be a 2-Sylow subgroup of the symmetric group $\mathrm{Sym}(2^n)$. Let also consider the cycle $\gamma_n = (1, \ldots, 2^n)$ of order $2^n$.
If we assume moreover that ...

**-2**

votes

**1**answer

222 views

### A generalization of an old group problem [closed]

Here is an old exercise in group theory: (1) If $G$ is a group of order $2n$ with $n$ odd then $G$ is not simple and in fact $G$ has a normal subgroup of order $n$. I am going for one straight ...

**5**

votes

**0**answers

336 views

### 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 ...

**6**

votes

**3**answers

595 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 ...

**7**

votes

**1**answer

433 views

### Littlewood Richardson rule and seminormal basis of Specht modules

Background
Seminormal Basis of Specht modules of $\mathfrak{S}_n$
Let $\lambda$ be a partition of $n$. A $\lambda$-tableau is a
bijection $\mathfrak{t}:\lambda \to \{1,2,...,n\}$. We say a ...

**7**

votes

**1**answer

601 views

### 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 ...

**4**

votes

**1**answer

619 views

### 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} ...

**3**

votes

**1**answer

197 views

### Combinatorics of index sets multiplicities in characters of symmetric groups

Hi everyone.
I'm pondering the following question: I have a Coxeter group $(W,S)$ of type $A_{n-1}$, i.e. the symmetric group $W=Sym(n)$ with the neighbour transpositions as generating set $S=\lbrace ...