All Questions
Tagged with partitions rt.representation-theory
31 questions
27
votes
3
answers
2k
views
Majorization and Schur Polynomials
Let me first define the majorization order (or dominance order) on partitions as $\lambda \succeq \mu$ iff $$\sum _{i=1}^{k}\lambda_i \geq \sum_{i=1}^{k}\mu_i$$ for all $1\le k\le l-1$ and $$\lambda_1+...
- 85.6k
14
votes
2
answers
571
views
Number of d-Calabi-Yau partitions
This problem arises from algebraic geometry/representation theory, see https://arxiv.org/pdf/1409.0668.pdf (chapter 2).
We call a partition $p=[p_1,...,p_n]$ with $2 \leq p_1 \leq p_2 \leq ... \leq ...
- 26.5k
13
votes
1
answer
2k
views
Number of standard Young tableaux with fixed corner entry
For a partition $\lambda=(\lambda_1\geq \lambda_2\geq\ldots\geq \lambda_k)$ of $n$, let the set of standard Young tableau of shape $\lambda$ be denoted by $SYT(\lambda)$ with boxes at $(i,j)$ denoted ...
- 4,952
10
votes
1
answer
272
views
Plane partitions as irreducible representations
The irreducible representations of the symmetric group algebras $A_n=KS_n$ over a the complex numbers (or any field of characteristic 0) $K$ satisfy the following properties:
The irreducible ...
- 26.5k
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
535
views
An identity related to partitions into $n$ parts and Schur polynomials
While working with Schur polynomials I found what seems like a nice identity, and I wonder if it has a simple proof.
Notation: Suppose $d,n\in\mathbb{N}$, and $\lambda =(\lambda_1,\dots,\lambda_n)$ ...
- 251
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
8
votes
0
answers
318
views
Why are these Littlewood-Richardson coefficients congruent to 1 mod 8?
Let $n\in{\mathbb N}$ and write $n=q_1+q_2+\dots+q_t$, where $q_1>q_2>\dots>q_t$ are powers of $2$. Let $\lambda_n$ be the partition with Frobenius symbol $(q_1-1,q_2-1,\dots,q_t-1;q_t,q_{t-1}...
- 1,090
7
votes
1
answer
272
views
SYT and contents of a partition
Let $\lambda$ be an integer partition, denote the number of Standard Young Tableaux of shape $\lambda$ by $f_{\lambda}$. This number is computed by the formula
$$f_{\lambda}=\frac{n!}{\prod_{u\in\...
- 43.2k
7
votes
1
answer
582
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
1
answer
444
views
A binary hook-length formula?
This is purely exploratory and inspired by curiosity.
Setup: For an integer $k>0$, let $k=\sum_{j\geq0}k_j2^j$ be its binary expansion and denote the sum of its digits by $\eta(k):=\sum_jk_j$. ...
- 43.2k
6
votes
1
answer
186
views
Action of $\widehat{\mathfrak{sl}_2}$ on symmetric functions with $\mathbb{Z}_{(2)}$ coefficients
It is known that there is a representation of the affine Lie algebra $\widehat{\mathfrak{sl}_q}$ (over $\mathbb{Z}$) on the algebra of symmetric functions, where the action of the Chevalley generators ...
- 3,446
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
5
votes
0
answers
357
views
$\text{Determinant}=(\sum \text{Determinant})^2$
Denote by $\delta_{n-1}=(n-1,n-2,\dots,1,0,0,\dots)$ the staircase partition and the embedded partition
$\lambda=(\lambda_1,\lambda_2,\dots)\subset\delta_{n-1}$.
QUESTION 1. Is this true?
$$\det\...
- 43.2k
5
votes
0
answers
131
views
Identity for classes of plane partitions
There are several classes of plane partitions in the literature.
Among these, let's look at the enumeration of three of them: the symmetric (SPP), totally symmetric (TSPP) and totally symmetric and ...
- 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
4
votes
0
answers
205
views
Dimension of a certain space of symmetric functions: Part I
Let $s_{\lambda}$ denote the Schur polynomial associated to a partition $\lambda$. A partition $\lambda$ is called a $t$-core if none of its hook lengths are multiples of $t$.
QUESTION. Consider the ...
- 43.2k
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
73
views
While expanding Jack polynomials in monomial basis
Denote $\mathbf{z}=(z_1,\dots,z_n)$. Let $P_{\kappa}(\mathbf{z};\alpha)$ be the symmetric Jack polynomials and suppose they are expanded in terms of the monomial symmetric basis $m_{\rho}(\mathbf{z})$ ...
- 43.2k
3
votes
0
answers
312
views
Enumerating a class of polynomials
How many equivalence classes of $\Bbb F_2[x,y]$ polynomials with $x$ degree $n_x$ and $y$ degree $n_y$ are there such that each $y^i$ coefficient (polynomial in $\Bbb Z[x]$) is distinct and $x^i$ ...
user76479
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
228
views
Ramanujan's theta functions and hook lengths?
Given an integer partition $\lambda\vdash n$ of $n$, one may associate a Young diagram $Y(\lambda)$ to it followed by a computation of hook length $h_{\square}$ for each cell $\square=(i,j)$ in $Y(\...
- 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....
2
votes
0
answers
223
views
Explicit formulas for the action of the Hall algebra of the cyclic quiver on q-Fock space?
In their paper on the decomposition numbers of Schur algebra, Vasserot and Varagnolo introduce an action of the (twisted) Hall algebra of a cyclic quiver $\Gamma$ on q-Fock space.
Without q-shifts, ...
- 44.7k
1
vote
0
answers
90
views
Dimension of a certain space of symmetric functions: Part II
This is the second installment of my earlier MO question.
Let $s_{\lambda}$ denote the Schur polynomial associated to a partition $\lambda$. Denote the set of all partitions with distinct parts by $\...
- 43.2k