All Questions
Tagged with symmetric-groups algebraic-combinatorics
18 questions
8
votes
1
answer
356
views
Describing the hook part of the symmetric group algebra
Let $\mathbf{k}$ be a field of characteristic $0$. Let $n\in\mathbb{N}$, and
consider the symmetric group $S_{n}$ consisting of all permutations of
$\left[ n\right] :=\left\{ 1,2,\ldots,n\right\} $...
- 33.8k
15
votes
1
answer
639
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 $\...
- 492
2
votes
0
answers
283
views
Geometric or combinatorial interpretations of the (weak) Bruhat order?
$\DeclareMathOperator\Inv{Inv}$The weak Bruhat order on the symmetric group has a straightforward combinatorial interpretation: Consider a set of labelled balls $1,2,\dotsc,n$. Then for two ...
- 232
8
votes
1
answer
272
views
Branching rule for Specht modules over Kazhdan-Lusztig basis
Suppose $\lambda\vdash n$ is a partition and $S^\lambda$ is the associated irreducible representation of $S_n$. As we know from the branching rule, we have an isomorphism of $S_{n-1}$ modules
$$S^\...
- 221
7
votes
1
answer
591
views
Can Matsumoto's theorem for the symmetric group be proved using a monovariant?
This is a question that can be asked for any Coxeter group, but for the sake of simplicity I will restrict myself to symmetric groups. Recall the main definitions:
Let $n$ be a nonnegative integer. ...
- 33.8k
9
votes
2
answers
572
views
Decomposing a polynomial ring into Specht Modules
Let $S_{\pi}$ where $\pi$ is an integer partition of $n$, denote the Specht module corresponding to $\pi$.
I am trying to decompose the set of all homogeneous polynomials in $x_1,x_2,...,x_n$ ...
- 261
1
vote
0
answers
177
views
Combinatorial bijection on monotone sequences
Let $(n),\mu$ be the partition of $n$ define $H_g^{m}((n);\mu)$ count's the number of tuples $(\tau_1,\ldots,\tau_r)$ of transposition in symmetric group $S_n$ with the following conditions
$$ (1,2,\...
- 685
5
votes
1
answer
290
views
Toggles for non-broken-circuit sets in matroids
Let $M$ be a matroid with ground set $E$. If $t$ is a total order on $E$, and if $S$ is a nonempty subset of $E$, then $\max_t S$ will mean the $t$-largest element of $S$ (that is, the maximum of $S$ ...
- 33.8k
13
votes
3
answers
678
views
Is this sum of cycles invertible in $\mathbb QS_n$?
I am interested the following element of the group algebra $\mathbb{Q}S_n$:
\begin{align}
\phi_n=2e+(1\ 2)+(1\ 2\ 3)+\dotsb+(1\ldots n)
\end{align}
where $e$ is the identity permutation. My question ...
- 380
5
votes
0
answers
267
views
Tabloid Construction of Permutation Representation of Hyperoctahedral Group
For a partition $\lambda \vdash n$, the permutation representation $M^{\lambda}$ of the symmetric group can be constructed in two ways. First, it may be written as the induced representation $M^{\...
- 297
49
votes
4
answers
6k
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\...
- 33.8k
0
votes
1
answer
232
views
Number of Boolean algebra subintervals in weak order of $S_n$
I'm wondering if anybody has an easy way to compute the number of subintervals in weak order of $S_n$ (considered as a Coxeter group of type $A_{n-1}$) that are isomorphic to Boolean algebras. I know $...
- 2,168
4
votes
1
answer
195
views
bijection between S-modules and Schur functors
Given a $\mathbb{S}$-module, we can construct a Schur functor, which is an endo functor in the category of vectorspaces. But given a Schur functor, how can we get back the $\mathbb{S}$-module? In ...
- 101
24
votes
0
answers
894
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-...
- 609
26
votes
6
answers
3k
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 ...
- 33.8k
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\...
- 33.8k
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, ...
- 33.8k
11
votes
1
answer
887
views
Dual of a Specht module
For a partition $\mu$ of $n$, let $S^{\mu}$ be the associated Specht module, defined over $\mathbb{Z}$. For any field $k$, we can tensor $S^{\mu}$ with $k$ to get a representation $S^{\mu}_k$ of the ...
- 113