All Questions
Tagged with young-tableaux symmetric-functions
12
questions
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32
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Linear involution for Specht modules
Let $n$ be a positive integer and $\lambda$ be a partition of $n$, which we identify with its Young diagram. Let $S^{\lambda}$ be the Specht module associated to $\lambda$.
Here the Specht modules are ...
- 3,052
1
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0
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92
views
Sum involving ${\frak{S}}_n$-character values and Kostka numbers
Let $\lambda$, $\mu$, and $\rho$ be partitions of $n$ and let
$\chi^\lambda_\rho$ and $K_{\lambda \mu}$ denote the associated ${\frak{S}}_n$-character value and Kostka number respectively.
Question: ...
- 567
2
votes
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answers
106
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A question about Fomin's local rules for growth diagrams
Let $w\in S_n$. Define the growth diagram of $w$ as follows: Start by an array of $n\times n$ squares, with an $X$ in the i'th column and row $w(i)$ from bottom. Then we obtain $(n+1)^2$ vertices (the ...
- 3,052
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123
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How to evaluate $s_\lambda(q,q^2,\cdots,q^m)$? (principal specialisation of the schur function)
It is required to show that
$$
s_\lambda(q,q^2,\cdots,q^m) = q^{m(\lambda)}\prod_{i,j \in \lambda}\frac{1-q^{c_{i,j}+m}}{1-q^{h_{i,j}}}
$$
where $c_{i,j}=j-i$ is the content of cell $(i,j)$, and $h_{i,...
user614535
3
votes
0
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317
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Jacobi–Trudi Determinants — how to use the Lindström–Gessel–Viennot Lemma to prove the second identity?
I am reading Bruce Sagan's Combinatorics: The Art of Counting. In $\S$7.2 The Schur Basis of $\mathrm{Sym}$, the author states the formulas involving the Jacobi–Trudi determinants and the Schur ...
user279515
2
votes
1
answer
94
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LR-rule and Standard Young Tableau counting
given that $s_\lambda s_\mu=\sum_{\nu} C_{\lambda \mu}^\nu s_\nu$ with $\vert \lambda\vert +\vert\mu \vert=\vert \nu\vert$, why does apparently also hold that $$h(\lambda) h(\mu) {\vert \nu\vert \...
- 910
1
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136
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Hook-content polynomial
Let $\lambda$ denote the hook of size $d$ for a fixed integer $d>0$. For $d=2$ there are two kinds of hooks for $d=3$ there are 3 different kind and so on. $c(\Box)$ denote the content of the $\...
- 1,065
1
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0
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315
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Polynomials invariant under specific permutations or subgroups of $S_n$.
The Schur polynomials or the power symmetric polynomials are such that they are invariant under the whole $S_n$. Are there polynomials which are invariant only under a chosen permutation group element ...
- 389
2
votes
1
answer
119
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(representation theoretic) meaning of sum over even rows of a Young tableau
Think of a Young tableau $R$ as composed by
$d$ rows with number of elements $\mu_i:=\mu_i^R$
$\mu_1 \geq \mu_2 \geq \cdots \geq \mu_d > \mu_{d+1}=0$
(and $\mu_i =0\, \forall i >d$)
and define
$\...
- 2,390
1
vote
2
answers
1k
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Combinatorial definition of Hall–Littlewood polynomials (sum over SSYT?)
Hall–Littlewood polynomials $P_\lambda(x;t)$ is an important deformation of Schur polynomials
forming a basis in the ring of symmetric polynomials over $\mathbb Z[t]$.
There are various definitions, ...
- 17.6k
3
votes
0
answers
247
views
Cauchy Identity for a specialized product of Schur polynomials
Let $\lambda=(\lambda_1,\lambda_2,\cdots,\lambda_d)$ be a partition, with $|\lambda|=n$. Let $\nu=\nu(\lambda):=(\lambda_1-1,\lambda_2,\cdots,\lambda_d).$ In other words, $\nu$ is obtained from $\...
- 32.9k
1
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0
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188
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Dimension of the Image of Young Projectors corresponding to Tensor factors.
Suppose I define the action of the symmetric group on abstract tensors as shuffling indices. I know this is very naive. I apologise, I am a physicist and working on a problem that involves tensors ...
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