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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Compute Schutzenberger involution of a Young tableau without using Jeu de Taquin

How does one compute Schutezenberger involution $T'$ of a Young tableau $T$ without using Jeu de Taquin. Can we use Viennot's construction or some other technique and apply it on the contents of $T$ ...
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Question on definition of Schur polynomial from Fulton's Young Tableaux.

In the following from Young Tableaux by Fulton, what happens if our Tableaux has numbers greater than $m$? Fulton gives an example with $m=6$, but according to his definition, we also have a monomial ...
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Showing that a tensor is zero through Young tableaux

There is a proof in Krupka: The Total Divergence Equation involving Young tableaux that I don't fully understand. Ripping it from context, there is a tensor $$T^{I_1...I_m,j},$$where the indices ...
• 6,220
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Given n, do we have a formula for the greatest Hook number of an n-box Young diagram?

Obviously the least Hook number is 1, by considering boxes stacked vertically or horizontally. Is there a formula for the greatest possible Hook number ? EDIT: The Hook number of a Young diagram is ...
• 325
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Idempotents in the group algebra of Sn and Specht modules

I am studying the irreps of Sn. I will use the following example tableau: $$|1|2|\\ |3|4|$$ From what I understand, one approach to constructing the irreducible representations is as follows: For ...
• 325
1 vote
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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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• 135
1 vote
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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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Involutions, RSK, and content

Suppose I begin with an involution $\sigma$ in the symmetric group ${\frak{S}}_n$ written a product of disjoint transpositions $\sigma = (a_1, b_1) \cdots (a_k, b_k)$. By RSK we know that involutions ...
1 vote
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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 ...
• 1,875
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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 $$\#\{h_i\text{ even}\}-\#\{h_i\text{ odd}\}?$$ ...
• 400
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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 ...
• 549
1 vote
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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 ...
• 1,580
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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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Is A unsatisfiable if there is a completed tableau with all branches closed?

I am having troubles wrapping my head around unsatisfiability and satisfiability. I understand that A is said to be satisfiable if there exists at least one case where the formula A is true. But when ...
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1 vote
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RSK and Matrices

It is well known that the RSK algorithm assigns to every square matrix with nonnegative integer entries a pair of semistandard Young Tableaux of same shape. The matrices are here used as just a square ...
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1 vote
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Expansion of polytabloids in the standard basis

I would like to know the most efficient way to write a polytabloid in terms of standard ones. I know the Garnir elements, but using them to do calculations is hard. I also read about "quadratic ...
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