# 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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### 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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### 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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### 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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### The symmetric group acting on $F_q^{\otimes n}$

Let $V$ be a complex vector space with dimension $dim(V) = k$, and let $\lambda = (\lambda_1, \dots, \lambda_d)$ be a partition of $n$. It is known (Fulton & Harris p. 86) that if $c_\lambda$ is ...
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### Reference for Identities of Young Symmetrizers

Considering Young Tableaux filled with numbers $1,...,n$ in a natural way, (left-to-right, row-by-row), the book "Representation Theory" of Fulton and Harris (Exercise 4.24) states that for all $x$ in ...
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### GAP: how to obtain the Young Symmetrizer?

Given a partition $\lambda$ of $n$ and a standard Young Tableaux filled with numbers from $1$ to $n$ (e.g. increasing row by row), how does one obtain the corresponding Young Symmetrizer using GAP? ...
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### Cycle type of a permutation in $S_n$ and its relation to partition of $n$ and its Young diagram

I know that it's possible to assign to each permutation its cycle type. I found two definitions of the cycle type and its relation to a partition of $n$: First definition Given $\sigma \in S_n$ ...
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### Smallest rectangle inscribing Young tableaux

I am interested in knowing the name of any of this characteristics of a Young tableaux (Ferrer's diagram): Smallest rectangle that contains it or the area of such a rectangle. For instance: Partition ...
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### Bijection between Standard Young Tableaux of height $\leq 2$ and $\lfloor n/2 \rfloor$-element subsets of $[n]$.

In OEIS sequence A001405, Mike Zabrocki claims that the number of Standard Young Tableaux of length $\leq 2$ is equal to $\binom{n}{\lfloor n/2 \rfloor}$. I haven't been able to conjure up a ...
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### Complex conjugated representation and Young tableaux

Imagine you have the Young tableu and the Dynkin numbers, $(q_1, q_2, ..., q_r)$, of the Lie algebra of $SU(n)$ which has $r$ simple roots. The way I assign Dynkin numbers is increasing its value from ...
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### Hook-length formula (equivalent)

I would like to know if anyone knows where I can find a proof for the equivalent hook-length formula $$f^{\lambda}=\frac{n! \cdot \prod_{i<j}(l_i-l_j)}{l_1! \cdot l_2! \cdot ... \cdot l_k!}$$ ...
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### Irreducible representations of symmetric group

Thanks to characters of representation we know that exists a bijection between irreducible representations of a finite group G and its conjugate classes. That bijection is proved showing that the ...
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### LR-rule and Standard Young Tableau counting

given that $s_\lambda s_\mu=\sum_{\nu} C_{\lambda \mu}^\nu s_\nu$ with $\vert \lambda\vert +\vert\mu \vert=\vert \nu\vert$, why does apparently also hold that h(\lambda) h(\mu) {\vert \nu\vert \...
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### Schur-Weyl duality for qubits

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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### $\mu$, $\nu$ are compositions with the same parts then for any $\lambda$, $K_{\lambda\mu}=K_{\lambda\nu}$ ($K$ Kostka number)

I want to show the following. If $\mu, \nu$ are compositions with the same parts (only rearranged) then for any $\lambda$ we have that $K_{\lambda\mu}=K_{\lambda\nu}$. I know that the Kostka ...
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### Proof that the Lascoux-Schützenberger involutions satisfies the braid-relations

I am interested in the Lascoux-Schützenbereger involutions $\theta_i$, defined on semistandard Young tableaux. See this paper for definitions, page 4. These involutions satisfy the braid relations: (...
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### Irreducible decomposition of Lorentz tensors with Young Tableaux

I want to understand the irreducible decomposition of Lorentz tensors by using Young tableaux. Let me start with a trivial example. Suppose we work in $n=4$ dimensions, and that we have a rank 2 ...
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### Question from Knuth problem on primitive networks and Young Tableaux of a particular shape.

It seems like from here that chains in the brahut in $S_n$ order are length $\binom{n}{2}$ and there is a bijection with young tableaux of shape $\lambda = (n-1, n-2, \ldots,1)$. Knuth in problem 38 ...
### Does $\mathbf{n}\otimes \mathbf{m}=\mathbf{m}\otimes \mathbf{n}$ in the Clebsch-Gordan decomposition?
Consider the two irreducible representations, $\mathbf{n}$ and $\mathbf{m}$, of $su_\Bbb{C}(N)$. The tensor product $\mathbf{n}\otimes \mathbf{m}$ can be found using Young tableaux as follows: Write \$...