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.
192
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Schur functors = Weyl functors in characteristic zero?
In the paper `Schur functors and Schur complexes' by Akin et al., the notion of a Schur functor had been defined for the first time over an arbitrary commutative ring $R$.
To recall the definition (I ...
- 576
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89
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Dimension of irreducible totally-symmetric tensor product of two su(n) representations
Consider the irreducible representation of $\mathfrak{su}$(n) given by the Young tableu
The dimension of the representation is easily obtained by the usual rules, and it is $d=n(n-1)/2$. One can also ...
- 133
2
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106
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Inequality regarding kostka numbers in representation theory
Before I post my question, let me set up some notation.
Notation. For $k\geq 1$, let $\lambda \vdash k$ be a partition of $[k]$. Let $C(k,m)$ be the set of all partitions $\lambda \vdash k$ of size $m$...
- 21
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0
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33
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Tableau which corresponds to alternating square representation
Recall that $S_5$ acts on $\mathbb C^5$ by permuting its coordinates and that $\mathbb C^5$ decomposes as $V\oplus W$ where $W$ is the trivial representation and $V$ has dimension $4$. Show that the ...
- 8,425
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1
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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
$$\...
- 8,425
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105
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Idempotency of the Young symmetrizer
Let $t$ be a (not necessarily standard) tableau of shape $\lambda \vdash n$ consisting of the numbers $1, ..., n$, each used exactly once.
Notation:
(i) $\mathfrak{S}_{n}$ is the symmetric group on $n$...
- 235
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1
answer
92
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Permuting the rows in ascending order first and then the columns of any Young tableau gives a standard Young tableau
Show that if you take any Young tableau and permute the rows in ascending order first and then the columns in ascending order (or columns first and then row), then you get a standard Young tableau.
I ...
- 8,425
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23
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About using Young tableaux to find non isomorphic trees on N vertices
In these days I dive in combinatorics to estimate the class of non isomorphic trees on N vertices.
After a while I realized this sounds me really similar when using Young Tableaux on permutation ...
2
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0
answers
71
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Weighted sum over integer partitions involving hook lengths
I am trying to compute the following quantity:
$$ 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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Matrix-Ball Construction - Young Tableaux
In William Fulton's Young Tableaux, Chapter $4$ on the Robinson-Schensted-Knuth Theorem outlines the matrix-ball construction. This allows us to begin with a matrix $A$ with nonnegative integer ...
- 115
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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^{...
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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
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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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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
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0
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23
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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 ...
- 2,572
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91
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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 ...
- 103
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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 ...
- 4,032
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68
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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 $V$, ...
- 469
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1
answer
98
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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
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138
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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
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0
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86
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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 ...
- 11.9k
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2
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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 ...
- 433
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5
answers
514
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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 ...
- 4,029
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1
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138
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Idempotents in the group algebra of Sn and 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 ...
- 433
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vote
0
answers
223
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Irreducible representations of $S_5$ and their Young diagrams
Given a Young tableau, we can construct its Young symmetrizer $c_\lambda$. Then, the ideal $\mathbb{C} S_n \cdot c_\lambda$ is an irreducible representation of $S_n$. Exercise 4.5 in Fulton and Harris ...
- 162
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132
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About Young symmetrizer $c_{\lambda}$
I'm reading the Fulton and Harris's book "Representation Theory". I want to ask about the proof of lemma 4.25.
Let $c_{\lambda}$ be the young symmetrizer, and let $V_{\lambda} = {\mathbb C}...
- 51
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0
answers
84
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Young-Tableau Exercise Solution Help
i have the following definition of a young tableau:
A Young tableau is an m × n matrix ($t_{i,j}$) with entries from N ∪ {∞}, for which it holds that in each row and each column the values ascend from ...
- 41
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0
answers
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Scalars by which symmetrizations of cyclic permutations act on Specht modules
Let $S_n$ be the symmetric group. Pick $a \in 2,\ldots,n$ and denote by $c_a \in \mathbb{C}[S_n]$ the symmetrization of the element $(12\ldots a)$ i.e. $c_a$ is the sum of cycles of type $a$.
Let $\...
- 135
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0
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84
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Sum involving ${\frak{S}}_n$-character values and Kostka numbers
Let $\lambda$, $\mu$, and $\rho$ be partitions of $n$ and let
$\chi^\lambda_\rho$ and $K_{\lambda \mu}$ denote the associated ${\frak{S}}_n$-character value and Kostka number respectively.
Question: ...
- 557
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0
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A question about Fomin's local rules for growth diagrams
Let $w\in S_n$. Define the growth diagram of $w$ as follows: Start by an array of $n\times n$ squares, with an $X$ in the i'th column and row $w(i)$ from bottom. Then we obtain $(n+1)^2$ vertices (the ...
- 2,572
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1
answer
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Parity of hooklengths in a partition diagram
Main Question
Let $\lambda\vdash n$ be a partition, with hooklengths $\{h_1,\dots,h_n\}$ in its partition diagram. Is there a formula for determining
$$\#\{h_i\text{ even}\}-\#\{h_i\text{ odd}\}?$$
...
- 405
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0
answers
169
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Branching rule for $S_n$ proof by James
Apologies for my English in advanced..
The following is a part from James' proof for the branching rule on the symmetric group:
It can be found in "The Representation Theory of the Symmetric ...
- 549
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0
answers
49
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Character of the irreducible representation $ψ^λ$ : $S_4$ → $Aut_C(S^λ)$
I am struggling with these exercise from group representations and would really appreciate some steps to take or sources with similar exercises.
The task is to compute the character of the irreducible ...
- 2,062
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1
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88
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Promotion on semistandard Young tableaux.
I searched on google and found that algorithms describe promotion operator on the set of standard Young tableaux. For example, the article. But I didn't find algorithms describe promotion operator on ...
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1
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49
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Computing permutation character associated to a Young subgroup.
If $\lambda = (\lambda_1,\lambda_2,\ldots)$ is a partition of $n$, then there is a permutation character of $S_n$ associated to the Young subgroup $S_\lambda$:
$$
\pi_\lambda = \mathrm{Ind}_{S_\lambda}...
- 702
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0
answers
41
views
Does the product of two Schur functions always have a lattice structure with respect to the dominance order of partitions?
The product of two Schur functions can be decomposed into a linear combination of other Schur functions according to the Littlewood-Richardson rule. This is also how the irreducible representations in ...
- 49
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1
answer
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Is a Standard Tableau determined by its descent set?
Suppose $\lambda\vdash n$ is a partition. Associated with this partition is the set of Standard Young Tableau $\text{SYT}(\lambda)$ such that the associated Young Diagram is filled in with the numbers ...
- 405
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1
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A question on Young tableau.
I am reading Fulton's book representation theory. My question occurs in the proof of Lemma 4.23. I will introduce my question concisely without letting you read that book.
The book introduces an order:...
- 1,352
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How to evaluate $s_\lambda(q,q^2,\cdots,q^m)$? (principal specialisation of the schur function)
It is required to show that
$$
s_\lambda(q,q^2,\cdots,q^m) = q^{m(\lambda)}\prod_{i,j \in \lambda}\frac{1-q^{c_{i,j}+m}}{1-q^{h_{i,j}}}
$$
where $c_{i,j}=j-i$ is the content of cell $(i,j)$, and $h_{i,...
user614535
0
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73
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Partition of integer and its conjugate
For the partition $(6,4,4,2)$ of integer $16$, if we draw its Young diagram with four rows of boxes, one below the other, of size $6$, $4$, $4$, and $2$, then flipping the resulting Young diagram ...
- 2,777
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49
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Combinatorics and Catalan Numbers [duplicate]
I was asked to investigate this question and to present my findings and I would like some sense of help and direction, I am very lost:-(
2n people, all of different heights
How many ways are there ...
1
vote
0
answers
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Embed U(5) to U(16) by specifying the 16-dimensional complex representation
$\DeclareMathOperator\SU{SU}\DeclareMathOperator\U{U}\DeclareMathOperator\Spin{Spin}$
My question concerns all the possible ways to embed SU(5) to U(16) by specifying the 16-dimensional complex ...
- 5,849
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0
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49
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Proof of conjecture about orthogonalized Specht polynomials.
Let $\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_r)$ be a $r$-part partition of integer $N$ ($\lambda\vdash N$), i.e.,
$$\sum_{i=1}^r{\lambda_i}=N,$$
such that $\lambda_i\leq\lambda_j$ for $r\geq j>...
- 155
2
votes
1
answer
76
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Weyl constructions for finite groups
Let $G$ be a finite group.
Is there a complex finite dimensional irreducible representation $V$ such that all irreducible ones are submodules of $V^{\otimes n}$ for some $n \in \mathbb{N}$?
If not, ...
- 1,782
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votes
0
answers
16
views
Number of building block structures following a set of rules
If we have an n amount of building blocks, and we are tasked with making, what I have found out are 'Young Tableau' like structures (though I am unsure if they are exactly the same), how many can we ...
4
votes
1
answer
212
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Operators commuting with tensor product representations of SU(2)
I am currently investigating $SU(2)$ symmetric qubit systems. In the course of this work I proved the following theorem:
Let $S_n$ denote the permutation group of $n$ elements. For $\sigma\in S_n$ ...
- 125
1
vote
1
answer
810
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$8 \otimes 8$ in $SU(3)$, dimension of the Young-tableau corresponding to the $\bar{10}$
In Georgi's Lie Algebras in Particle Physics, he calculates the decomposition of $8\otimes 8$ in $SU(3)$, and obtains
$$8\otimes 8 = 27 \oplus 10 \oplus \bar{10} \oplus 8 \oplus 8 \oplus 1,$$
...
- 147
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votes
0
answers
73
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Creation-Annihilation operators and Young diagrams
Let assume a Fock space written as,
$$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 ...
2
votes
1
answer
159
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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 ...
- 1,643
3
votes
1
answer
253
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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 ...
