Questions tagged [young-tableaux]
For questions on the Young tableau, a combinatorial object useful in representation theory and Schubert calculus. It provides a convenient way to describe the group representations of the symmetric and general linear groups and to study their properties.
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Irreducible representations of $S_5$ and their Young diagrams
Given a Young tableau, we can construct its Young symmetrizer $c_\lambda$. Then, the ideal $\mathbb{C} S_n \cdot c_\lambda$ is an irreducible representation of $S_n$. Exercise 4.5 in Fulton and Harris ...
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About Young symmetrizer $c_{\lambda}$
I'm reading the Fulton and Harris's book "Representation Theory". I want to ask about the proof of lemma 4.25.
Let $c_{\lambda}$ be the young symmetrizer, and let $V_{\lambda} = {\mathbb C}...
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Young-Tableau Exercise Solution Help
i have the following definition of a young tableau:
A Young tableau is an m × n matrix ($t_{i,j}$) with entries from N ∪ {∞}, for which it holds that in each row and each column the values ascend from ...
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Scalars by which symmetrizations of cyclic permutations act on Specht modules
Let $S_n$ be the symmetric group. Pick $a \in 2,\ldots,n$ and denote by $c_a \in \mathbb{C}[S_n]$ the symmetrization of the element $(12\ldots a)$ i.e. $c_a$ is the sum of cycles of type $a$.
Let $\...
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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: ...
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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 ...
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Parity of hooklengths in a partition diagram
Main Question
Let $\lambda\vdash n$ be a partition, with hooklengths $\{h_1,\dots,h_n\}$ in its partition diagram. Is there a formula for determining
$$\#\{h_i\text{ even}\}-\#\{h_i\text{ odd}\}?$$
...
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Branching rule for $S_n$ proof by James
Apologies for my English in advanced..
The following is a part from James' proof for the branching rule on the symmetric group:
It can be found in "The Representation Theory of the Symmetric ...
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Character of the irreducible representation $ψ^λ$ : $S_4$ → $Aut_C(S^λ)$
I am struggling with these exercise from group representations and would really appreciate some steps to take or sources with similar exercises.
The task is to compute the character of the irreducible ...
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Promotion on semistandard Young tableaux.
I searched on google and found that algorithms describe promotion operator on the set of standard Young tableaux. For example, the article. But I didn't find algorithms describe promotion operator on ...
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Computing permutation character associated to a Young subgroup.
If $\lambda = (\lambda_1,\lambda_2,\ldots)$ is a partition of $n$, then there is a permutation character of $S_n$ associated to the Young subgroup $S_\lambda$:
$$
\pi_\lambda = \mathrm{Ind}_{S_\lambda}...
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Does the product of two Schur functions always have a lattice structure with respect to the dominance order of partitions?
The product of two Schur functions can be decomposed into a linear combination of other Schur functions according to the Littlewood-Richardson rule. This is also how the irreducible representations in ...
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Is a Standard Tableau determined by its descent set?
Suppose $\lambda\vdash n$ is a partition. Associated with this partition is the set of Standard Young Tableau $\text{SYT}(\lambda)$ such that the associated Young Diagram is filled in with the numbers ...
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A question on Young tableau.
I am reading Fulton's book representation theory. My question occurs in the proof of Lemma 4.23. I will introduce my question concisely without letting you read that book.
The book introduces an order:...
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How to evaluate $s_\lambda(q,q^2,\cdots,q^m)$? (principal specialisation of the schur function)
It is required to show that
$$
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,...
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