**8**

votes

**0**answers

83 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

33 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

57 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 ...

**2**

votes

**0**answers

99 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

447 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

317 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 ...

**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

50 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 ...

**4**

votes

**2**answers

160 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 ...

**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 ...

**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 ...

**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

**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 ...

**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

159 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 ...

**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.
$n=1,...,6$, ...

**2**

votes

**2**answers

199 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 ...

**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
...

**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 ...

**9**

votes

**0**answers

393 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

340 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

559 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 ...

**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 ...

**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 ...

**6**

votes

**2**answers

525 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 ...

**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 ...

**-2**

votes

**1**answer

221 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 ...

**13**

votes

**2**answers

842 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 ...

**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 ...

**16**

votes

**1**answer

651 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, ...

**7**

votes

**1**answer

597 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 ...

**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 ...

**2**

votes

**2**answers

285 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 ...

**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 ...

**1**

vote

**0**answers

75 views

### On generalization of Wigner semi circle

I want to analyse noise model for a matrix M whose entries are not real numbers. The matrix is a collection of N permutation matrices of size nxn i.e, M is NnxNn. Because its a collection of ...

**16**

votes

**1**answer

447 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. ...

**4**

votes

**1**answer

187 views

### Representations of Sym(n) and SL_d

Irreducible representations of the symmetric group Sym$(n)$, and degree-$n$ algebraic representations of SL$_d(\mathbb C)$ for $d\ge n$, can both be classified by Young diagrams with $n$ boxes.
...

**7**

votes

**1**answer

475 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 ...

**1**

vote

**0**answers

91 views

### Next smallest dimension of Specht Module after $(n)$, $(1^n)$, $(n-1,1)$ and $(2,1^{n-2})$

In representation theory of $S_n$, we know that for $n \geq 9$, the only Specht modules $S^\alpha$ of dimension $f^\alpha < {n-1 \choose 2} - 1$ are:
$S^{(n)}$ and $S^{(1^n)}$ with dimension $1$, ...

**5**

votes

**1**answer

241 views

### Dimension of Specht Modules $S^\lambda$

In the study of representation theory of $S_n$, we know that the irreducible characters of $\chi_\lambda$ of $S_n$ are indexed by partitions $\lambda \vdash n$. There are several methods in ...

**6**

votes

**1**answer

208 views

### Decomposition of $\mathrm{End}(V)$ as $S_n\times S_n$-module

Let $V$ be a finite-dimensional, complex vector space and set $\newcommand{\Gl}{\mathrm{Gl}}G:=\Gl(V)\times\Gl(V)$. Let $E:=\mathrm{End}(V)$ and consider its coordinate ring $\mathbb C[E]$, the space ...

**4**

votes

**2**answers

346 views

### A basis for Schur functors

Suppose $V$ is a finite-dimensional vector space (over $\mathbb{C}$) and $\lambda$ is a partition of $n$ (not necessarily the dimension). Let $S^\lambda(V)=(V^{\otimes n})_\lambda$ be the $\lambda$'th ...

**13**

votes

**4**answers

635 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 ...

**7**

votes

**1**answer

432 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 ...

**1**

vote

**1**answer

576 views

### Cyclic Subgroups of the Symmetric Group

If we write a partition $n=k_1+...+k_r$, then we can create a $(k_1,...,k_r)$-cycle in $S_n$ with order equal to the least common multiple of the $k_i$'s. It is clear that every cyclic subgroup will ...