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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Partition of integer and its conjugate
For the partition $(6,4,4,2)$ of integer $16$, if we draw its Young diagram with four rows of boxes, one below the other, of size $6$, $4$, $4$, and $2$, then flipping the resulting Young diagram ...
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Combinatorics and Catalan Numbers [duplicate]
I was asked to investigate this question and to present my findings and I would like some sense of help and direction, I am very lost:-(
2n people, all of different heights
How many ways are there ...
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Embed U(5) to U(16) by specifying the 16-dimensional complex representation
$\DeclareMathOperator\SU{SU}\DeclareMathOperator\U{U}\DeclareMathOperator\Spin{Spin}$
My question concerns all the possible ways to embed SU(5) to U(16) by specifying the 16-dimensional complex ...
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Proof of conjecture about orthogonalized Specht polynomials.
Let $\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_r)$ be a $r$-part partition of integer $N$ ($\lambda\vdash N$), i.e.,
$$\sum_{i=1}^r{\lambda_i}=N,$$
such that $\lambda_i\leq\lambda_j$ for $r\geq j>...
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Weyl constructions for finite groups
Let $G$ be a finite group.
Is there a complex finite dimensional irreducible representation $V$ such that all irreducible ones are submodules of $V^{\otimes n}$ for some $n \in \mathbb{N}$?
If not, ...
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Number of building block structures following a set of rules
If we have an n amount of building blocks, and we are tasked with making, what I have found out are 'Young Tableau' like structures (though I am unsure if they are exactly the same), how many can we ...
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Operators commuting with tensor product representations of SU(2)
I am currently investigating $SU(2)$ symmetric qubit systems. In the course of this work I proved the following theorem:
Let $S_n$ denote the permutation group of $n$ elements. For $\sigma\in S_n$ ...
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$8 \otimes 8$ in $SU(3)$, dimension of the Young-tableau corresponding to the $\bar{10}$
In Georgi's Lie Algebras in Particle Physics, he calculates the decomposition of $8\otimes 8$ in $SU(3)$, and obtains
$$8\otimes 8 = 27 \oplus 10 \oplus \bar{10} \oplus 8 \oplus 8 \oplus 1,$$
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Creation-Annihilation operators and Young diagrams
Let assume a Fock space written as,
$$F=\bigoplus_\rho V_\rho,$$
where $V_\rho$ is an irreducible representation of $U(N)$ labeled by a partition (Young diagram) $\rho$. For the so-called bosonic case ...
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Antisymmetric tensors of a tensor product: $\Lambda^k(V \otimes W)$
Given two vector spaces $V, W$ over $\mathbb{R}$, it's true that
$\Lambda^2 (V \otimes W) \cong \left(\Lambda^2 V \otimes S^2 W \right) \oplus \left( S^2 V \otimes \Lambda^2 W \right)$. If I'm seeing ...
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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 ...
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Young Tableaux for Lorentz Algebra Spinor Representations
I am new to this field and learning this subject on my own so apologies if I interpret anything incorrectly.
My question is while working out representation theory of semi-simple Lie groups through ...
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How many ways can a given Young tableau be written as part of the decomposition of tensor products of smaller ones?
Given a representation in $SU(N)$ with Young tableau $Y$, how many ways are there that I can write $$Y \in y_1\otimes y_2 \otimes \cdots \otimes y_n$$ where $n\leq |Y|$, the number of boxes there are ...
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Explicit construction of a representation of Young diagram/tableaux from fundamental representations
Given $SU(N)$ fundamental representation say $U^i$ in the fundamental $N$ of $SU(N)$, with indices $i=1,2,3,\dots,N$.
We can construct a representation whose Young diagram/tableaux look like
Given
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Distribution of number of terms in integer partitions
SOLVED: This is the Gumbel distribution
Let $\pi^n_i$ be set containing the terms in the $i$-th integer partition of the natural number $n$, according to whatever enumeration. For example, for $n = 5$ ...
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