All Questions
14 questions
10
votes
1
answer
274
views
When are immanants irreducible?
For a partition $\lambda$ let $\chi_\lambda$ be the corresponding irreducible representation of the symmetric group $S_n$.
Let $\mathrm{Imm}_\lambda(x) = \sum\limits_{\pi \in S_n} \chi_\lambda(\pi) x_{...
- 26.5k
8
votes
2
answers
742
views
A product identity for partitions
For a partition $\lambda=(\lambda_1\ge \lambda_2\ge \dots)$, let
$m_\lambda=\prod_i (\lambda_i-\lambda_{i+1})!$ be the product of factorials of consecutive differences and let $v_\lambda=\prod_{i | \...
- 7,952
7
votes
1
answer
583
views
Hurwitz numbers and $t$-cores
For integers $k \geq 0$ and $d \geq 1$ let $H(k,d)$
be the Hurwitz number which, for the purposes
of this posting, will be defined by:
\begin{equation}
H(k,d)
\, := \ d! \, \sum_{\lambda \, \vdash d}...
- 2,137
6
votes
1
answer
407
views
hooks and contents: Part I
For a cell $\square$ in the Young diagram of a partition $\lambda$, let $h_{\square}$ and $c_{\square}$ denote the hook length and content of $\square$, respectively.
R Stanley proved the following ...
- 43.2k
6
votes
0
answers
196
views
hooks and contents: Part II
This is a 2nd installment to my earlier MO question for which Mark Wildon furnished a clean answer.
$\mathcal{O}(\pi)$ and $\mathcal{E}(\pi)$ stand for the number of odd and even cycles of a ...
- 43.2k
4
votes
1
answer
324
views
What is the $p$-regular partition corresponding to the sign representation of $S_{n}$ over a field of characteristic $p$?
I'm now interested in the modular representation of symmetric groups.
It is well-known that for a fixed prime $p$, there is a bijection between the irreducible representations of $S_{n}$ over a field ...
- 1,013
4
votes
1
answer
528
views
The number of permutations of a given cycle type that fix a string with a given histogram
Let $\lambda$ and $\mu$ be partitions of some integer $n \geq 1$. Let $d$ be the number of parts in $\mu$ and let $\bar{\mu} \in \{1,\dotsc,d\}^n$ denote the string $1^{\mu_1} 2^{\mu_2} \dotsb d^{\...
- 499
4
votes
0
answers
313
views
What is $\dim D^{\lambda}$ for the symmetric group?
What are the dimensions of the simple modules $D^{\lambda}=S^{\lambda}/S^{\lambda}\cap (S^{\lambda})^{\perp}$ for the modular representation theory of $S_n$, i.e. $\operatorname{char}(k)=p>0$?
I ...
- 571
3
votes
1
answer
221
views
Asymptotics for number of $p$-regular partitions of $n$
The number of simple modules $D^{\lambda}=S^{\lambda}/S^{\lambda}\cap (S^{\lambda})^{\bot}$ of the symmetric group over a field $k$ such that $\text{char}(k)=p > 0$ is the number of $p$-regular ...
- 571
3
votes
2
answers
924
views
sum over all integer partitions, of the product of the factorials of the terms
I'm looking for something making tractable the sum, over all partitions into k terms of an integer n, of the product of the factorials of all the terms.
Thanks,
- 31
3
votes
0
answers
153
views
Adding a row to a Young Tableau via Novelli-Pak-Stoyanovskii
Let $T_{\lambda}$ be the set of standard young tableaux (SYT) of shape $\lambda_1\geq \lambda_2\cdots\geq \lambda_n$. Now consider pushing a row $\mu$ with $\mu\geq \lambda_1$ onto $Y$ to give shape $\...
- 4,952
2
votes
0
answers
352
views
On characters of the symmetric group: Part 1
Given an integer partition $\lambda$, denote $\ell(\lambda)=$ length, $\vert\lambda\vert=$ size and $\lambda=$ conjugate of $\lambda$. Allow to write $\lambda\vdash n$ either as $(\lambda_1,\dots,\...
- 43.2k
2
votes
0
answers
89
views
Upper bound on decomposition numbers for the symmetric group in a block of weight $w$
The $p$-blocks of the symmetric group $S_n$ are labelled by pairs $(\gamma, w)$ where $\gamma$ is a $p$-core partition, $w \in \mathbb{N}_0$ is the weight of the block, and $|\gamma| + p w = n$. ...
- 11.2k
2
votes
0
answers
161
views
Partitions limit shape and LDP
Hello!
I am trying to understand the paper of Dembo-Vershik-Zeitouni, Large deviations for integer partitions. I am only interested in Theorem 2, which deals with the case of the uniform distribution....