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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the number of Young tableaux in general
From the wiki page Catalan number, we know the number of Young tableaux whose diagram is a 2-by-n rectangle given $2n$ distinct numbers is $C_n$. In general, given $m\times n$ distinct numbers, how ...
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How to apply the Schur-Weyl duality to a three-qubit system?
I am interested in applying Schur-Weyl duality to quantum information theory, specifically "qubits". But I have been stuck for some time on understanding how the Young symmetrizers work in this ...
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Given $n$ distinct elements, how many Young Tableaux can you make?
Given $n$ distinct elements, how many Young Tableaux can one make?
user8583
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Direct proof of Gelfand-Zetlin identity
Denote by $D(a_1,\dots,a_n)$ the product $\prod_{j>i}(a_j-a_i)$. Assuming that $a_i$ are integers s.t. $a_1\le a_2\le\dots\le a_n$, prove that $D(a_1,...,a_n)/D(1,...,n)$ is the number of Gelfand-...
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What's the relation between standard Young tableaux and Catalan number?
From wikipedia, I know some basic facts about Catalan number and Young tableaux. Moreover, I know that Catalan number $C_n$ is the number of triangulations of a $n+2$-gon.
What's the relation ...
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The irreducible representation of $S_n$ corresponding to the partition $n = (n − 1) + 1$
Let a finite group $G$ act doubly transitively on a finite set $X$, i.e., given $x,y,z,w\in X$ such that $x\neq y$ and $z\neq w$, there is a $g\in G$ such that $g.x=z$ and $g.y=w$. So, we can write
$$\...
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Intuition behind Hook Length Formula
Given a nonnegative integer $n$. Show that the Catalan number $C_n$ is the number of ways to arrange the integers $1, 2, \ldots, 2n$ as a standard Young tableau of rectangular shape with dimensions $2 ...
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Show via Young diagrams that the standard representation of $S_d$ corresponds to the partition $d=(d-1)+1$
I'm working through Fulton-Harris and I'm kind of "stuck" at the following question. I'm looking for representations of $S_d$, the symmetric group on $d$ letters via Young Tableaux. The question is:
"...
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Young projectors in Fulton and Harris
In Section 4 of Fulton and Harris' book Representation Theory, they give the definition of a Young tableau of shape $\lambda = (\lambda_1,\dots,\lambda_k)$ and then define two subgroups of $S_d$, the ...
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Finding $\mathbf{10}\otimes \mathbf{8}\otimes \mathbf{8}\otimes \mathbf{8}$ in $SU(3)$
I know that in $SU(3)$
$$\mathbf{8}\otimes \mathbf{8} = \mathbf{27}+\mathbf{10}+\mathbf{\bar{10}}+\mathbf{8}+\mathbf{8}+\mathbf{1}. $$
How can one use this to compute $$\mathbf{10}\otimes \mathbf{8}\...
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Number of Standard Young Tableaux of $n$ cells
I know there is a $1-1$ correspondence between the number of standard young tableaux of $n$ cells and the number of involutions in $S_n$. Number of involutions in $S_n$ satisfies the recurrence ...
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Reference request: Representation theory over fields of characteristic zero
Many representation theory textbooks and online resources work with the field of complex numbers or more generally algebraically closed fields of characteristic zero. I am looking for a good textbook ...
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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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Young Tableaux Generalizing
The entries in an array include all the digits from 1 through 9, arranged so that the entries in every row and column are in increasing order. How many such arrays are there? (2010 AMC12 B)
The ...
user87611
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The number of permutations that can be written in two ways as a product of row and column permutations of a Young tableau
My question is related to an issue in the book "Young tableaux" by W.Fulton. Consider a Young tableau $T$ of a given fixed shape filled with integers $1,\ldots,n$. A permutation $\sigma$ in ...
