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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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 ...
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Decomposing $Res^{S_n}_{S_{n-1}}V_\lambda$
Let $\lambda$ be a partition of $n$. I'm trying to show that $Res^{S_n}_{S_{n-1}}V_\lambda\cong \oplus _{\mu:\mu \vdash\lambda}V_\mu$, ($\mu \vdash\lambda$ means that $\mu$ is a partition of $n-1$, ...
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Given a Ferrers diagram, prove that $\det(M)=1$
Let $\lambda$ be a Ferrers diagram corresponding to some
integer partition of $k$. We number the rows and the columns, so that the
j'th leftmost box in the i'th upmost row is denoted as $(i,j)$. Let
$...
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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
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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 ...
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Generating function of Young Diagram from a given semiperimeter
so my question is:
What is the generating function for the number of Young diagrams of a given semiperimeter?
My approach: knowing that there exists a diagram with zero boxes, $$a_0=1$$$$a_1=2$$ $$....
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Is A unsatisfiable if there is a completed tableau with all branches closed?
I am having troubles wrapping my head around unsatisfiability and satisfiability. I understand that A is said to be satisfiable if there exists at least one case where the formula A is true. But when ...
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RSK and Matrices
It is well known that the RSK algorithm assigns to every square matrix with nonnegative integer entries a pair of semistandard Young Tableaux of same shape. The matrices are here used as just a square ...
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Expansion of polytabloids in the standard basis
I would like to know the most efficient way to write a polytabloid in terms of standard ones.
I know the Garnir elements, but using them to do calculations is hard. I also read about "quadratic ...
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show that the number of standard tableau of shape $(n^2)$ is the Catalan number
How would one show that the number of standard tableau of shape $(n^2)$ is the Catalan number
$\mathrm{\frac{1}{n+1}}$$2n\choose{n}$
any help would be great.
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Maximum value, function of partition and its conjugate
Suppose that we have $n\in \mathbb{Z}_{+}$ and some $\alpha\ge 3$.
I am trying to find maximum value of:
$\sum_{i,j=1}^{n}|\lambda_{i}-\lambda_{j}^{*}|^{\alpha},$
over
$\{\lambda\in \mathbb{Z}^...
user617199
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Young tableaux - column group
I am studying young tableaux and at one point in a demonstration the author states that
$$C_{\pi t} = \pi C_{t}\pi^{-1}$$
where $C_{t}$ is a subgroup of $S_{n}$ consisting of permutations which only ...
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Which is the importance of Young’s tableaux in mathematics?
I don’t know much about combinatorics, I’m just getting started on this. I want to know, why Young’s tableaux are important? and why it is important to relate them to matrices?
Thank you very much.
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Putting Entries in Young Diagram to make Tableaux
I was reading the book on Young Tableaux by Fulton. On first page of notations, he defined Young diagram to be left justified rows of boxes, weakly decreasing downwards. Then, he defines Young ...
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Constructing a centrally primitive idempotent in the group algebra of the symmetric group
Consider the group algebra of the symmetric group $ \mathbb{C} S_k$.
Given some Young tableau $T$ of shape $\lambda$, let $a_{\lambda,T}$ and $b_{\lambda,T}$ be the row symmetrizer and column ...
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