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.
198
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Weighted sum over integer partitions involving hook lengths
I am trying to compute the following quantity:
$$ g_n(x) = \sum_{\lambda \vdash n} \prod_{h \in \mathcal{H}(\lambda)} \frac{1}{h^2} \exp\left[x\sum_i \binom{\lambda_i}{2} - \binom{\lambda_i'}{2}\right]...
- 484
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0
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34
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Constructing Young tableaux basic question
I'm trying to understand Young tableaux and was making some exercises.
I'm a bit confused with the following question
Given a tensor $B^{ijk}$ where $B^{ijk} = -B^{jik}$ and $B^{ijk} + B^{kij} + B^{...
- 796
2
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59
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Bijection between noncrossing sets of arcs and row-strict Young tableaux
There is a well-known bijection between (i) noncrossing partitions of the set {1,...,2n} where all blocks have size 2, and (ii) standard Young tableaux with n rows and 2 columns. There exist a Catalan ...
- 67
2
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1
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How many Young tableaux of size 6 are there? [duplicate]
How many Young tableaux of size 6 are there? I have come up with the number 76, by counting the number of every possible Young tableau of weight 6, namely
{6}, {5,1}, {4,2}, {4,1,1}, {3,3}, {3,2,1}, {...
- 502
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0
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49
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Number of semi-standard Young tableaux of shape $\lambda$ with some entries fixed
Given a partition $\lambda$, the number of semi-standard Young tableaux (SSYT) of shape $\lambda$ with maximum entry $n$ is given by
\begin{equation}
\prod_{1\leq i<j\leq n} \frac{\lambda_i-\...
- 11
1
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0
answers
32
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Linear involution for Specht modules
Let $n$ be a positive integer and $\lambda$ be a partition of $n$, which we identify with its Young diagram. Let $S^{\lambda}$ be the Specht module associated to $\lambda$.
Here the Specht modules are ...
- 3,052
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Negative powers of the determinant representation of $U(N)$
Consider the determinant representation of $U(N)$ defined by $\det:U(N)\ni U\mapsto\det U\in U(1)$. If I'm not mistaken, $\det^{n}$ for $n\geq 1$ are all irreducible representations. When classifying ...
- 113
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Littlewood-Richardson coefficients and conjugation of Young diagrams
I am currently reading William Fulton's Young Tableaux and struggling to understand the proof of Corollary 2 in Section 5 of the book.
Suppose that $\lambda$ and $\mu$ are Young diagrams (or ...
- 4,232
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Schur functors applied to irreducible representations of $S_n$
For a $d$-box Young diagram $\lambda$, the Schur functor is a functor $S_\lambda: \text{Vect}\rightarrow \text{Vect}$. If $\lambda = d$ then $S_\lambda V=S^d V$ the $d$-th symmetric power of $V$, ...
- 479
3
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Schur functors for $\mathfrak{S}_3$
I have been trying to calculate the explicit images of the Schur functors for the action of $\mathfrak{S}_3$ on $V^{\otimes 3}$ where $V$ is some vector space, for the sake of concreteness of ...
- 311
0
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0
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171
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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$ ...
- 99
0
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0
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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 ...
- 12.1k
3
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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 ...
- 443
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votes
5
answers
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What are the end and coend of Hom in Set?
A functor $F$ of the form $C^{op} \times C \to D$ may have an end $\int_c F(c, c)$ or a coend $\int^c F(c, c)$, as described for example in nLab or Categories for Programmers. I'm trying to get an ...
- 4,207
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Are idempotents in the group algebra of $S_n$ equivalent to 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 ...
- 443