All Questions
Tagged with young-tableaux symmetric-polynomials
12
questions
1
vote
0
answers
44
views
Does the product of two Schur functions always have a lattice structure with respect to the dominance order of partitions?
The product of two Schur functions can be decomposed into a linear combination of other Schur functions according to the Littlewood-Richardson rule. This is also how the irreducible representations in ...
- 59
1
vote
0
answers
80
views
On the value of a skew Schur function at the identity
The generating function $\frac{1}{(1-t)^N}=\sum_k {N+k-1\choose k}t^k=\sum_k h_k(1)t^k$ and the Jacobi-Trudi formula $s_{\lambda/\mu}=\det(h_{\lambda_i-i-\mu_j+j})$ tell me that the value of the skew ...
- 1,847
2
votes
0
answers
85
views
Combinatorics for exterior power for arbitrary Specht module
The exterior powers of the standard representation are easily seen to be the representations whose Young diagrams have only boxes in the first row or first column. But, what if I start with an ...
1
vote
0
answers
96
views
Murnaghan-Nakayama rule for general dimension of a hook
Let $m,r\in \mathbb{N}, n=rm$. The character of symmetric group $\chi^{\lambda}$ where $\lambda$ is of the from
$(k,1^{n-k})$ evaluated at the conjugacy class of $S_{n}$ of the from $(r,\ldots,r)$, ...
- 1,065
1
vote
0
answers
136
views
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
2
votes
1
answer
797
views
Schur function principal specialisation
Let $s_{\lambda}(p_1,p_2,)$ denote schur function in power-sum symetric basis. More precisely
$$ s_{\lambda}=\sum_{\mu}\chi_{\mu}^{\lambda}p_{\mu}|k_{\mu}|.(*)$$
$p_i=\sum_k x_k^i$.
For more ...
- 1,065
1
vote
0
answers
315
views
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
0
answers
549
views
Special case of Pieri-Rule
is there an "elementary" (read: short combinatorial) proof for the rule
$$ s_\lambda \cdot s_{(1)} = \sum_{\mu} s_{\mu} $$
where $\mu$ ranges over all partitions obtained from $\lambda$ by adding a ...
- 43
2
votes
1
answer
119
views
(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
6
votes
1
answer
196
views
Flattening Young Tableaux
Let $\lambda=(\lambda_1,\lambda_2,\cdots,\lambda_k)$ be a partition with $|\lambda|=n$ and $\lambda_1\geq \lambda_2\geq\cdots\geq \lambda_k$. For any Standard Young Tableaux (SYT) $T$ of shape $\...
- 32.9k
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
vote
0
answers
188
views
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 ...
- 63