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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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 $T$ ...
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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 ...
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Showing that a tensor is zero through Young tableaux

There is a proof in Krupka: The Total Divergence Equation involving Young tableaux that I don't fully understand. Ripping it from context, there is a tensor $$ T^{I_1...I_m,j}, $$where the indices ...
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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 ...
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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 ...
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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 ...
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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}...
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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 ...
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A counting problem with Young Diagrams

Let $n$ and $d$ be natural numbers. How many Young diagrams of size $n$ are there such that each diagram has less than or equal to $d$-rows ?
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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 $\...
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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: ...
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Involutions, RSK, and content

Suppose I begin with an involution $\sigma$ in the symmetric group ${\frak{S}}_n$ written a product of disjoint transpositions $\sigma = (a_1, b_1) \cdots (a_k, b_k)$. By RSK we know that involutions ...
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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 ...
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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 $$\#\{h_i\text{ even}\}-\#\{h_i\text{ odd}\}?$$ ...
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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 ...
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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 ...
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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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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}...
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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 ...
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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 ...
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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:...
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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,...
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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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3 votes
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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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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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Given a Ferrers diagram, prove that $\det(M)=1$

Let $\lambda$ be a Ferrers diagram corresponding to some integer partition of $k$. We number the rows and the columns, so that the j'th leftmost box in the i'th upmost row is denoted as $(i,j)$. Let $...
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Jacobi–Trudi Determinants — how to use the Lindström–Gessel–Viennot Lemma to prove the second identity?

I am reading Bruce Sagan's Combinatorics: The Art of Counting. In $\S$7.2 The Schur Basis of $\mathrm{Sym}$, the author states the formulas involving the Jacobi–Trudi determinants and the Schur ...
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Combinatorics for exterior power for arbitrary Specht module

The exterior powers of the standard representation are easily seen to be the representations whose Young diagrams have only boxes in the first row or first column. But, what if I start with an ...
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2 votes
1 answer
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Generating function of Young Diagram from a given semiperimeter

so my question is: What is the generating function for the number of Young diagrams of a given semiperimeter? My approach: knowing that there exists a diagram with zero boxes, $$a_0=1$$$$a_1=2$$ $$....
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Is A unsatisfiable if there is a completed tableau with all branches closed?

I am having troubles wrapping my head around unsatisfiability and satisfiability. I understand that A is said to be satisfiable if there exists at least one case where the formula A is true. But when ...
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RSK and Matrices

It is well known that the RSK algorithm assigns to every square matrix with nonnegative integer entries a pair of semistandard Young Tableaux of same shape. The matrices are here used as just a square ...
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Expansion of polytabloids in the standard basis

I would like to know the most efficient way to write a polytabloid in terms of standard ones. I know the Garnir elements, but using them to do calculations is hard. I also read about "quadratic ...
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show that the number of standard tableau of shape $(n^2)$ is the Catalan number

How would one show that the number of standard tableau of shape $(n^2)$ is the Catalan number $\mathrm{\frac{1}{n+1}}$$2n\choose{n}$ any help would be great.
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Maximum value, function of partition and its conjugate

Suppose that we have $n\in \mathbb{Z}_{+}$ and some $\alpha\ge 3$. I am trying to find maximum value of: $\sum_{i,j=1}^{n}|\lambda_{i}-\lambda_{j}^{*}|^{\alpha},$ over $\{\lambda\in \mathbb{Z}^...
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Young tableaux - column group

I am studying young tableaux and at one point in a demonstration the author states that $$C_{\pi t} = \pi C_{t}\pi^{-1}$$ where $C_{t}$ is a subgroup of $S_{n}$ consisting of permutations which only ...
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3 votes
3 answers
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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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