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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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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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1answer
52 views

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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113 views

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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1answer
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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 ...
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Ordering on $\lambda$-tabloids

Let $\lambda$ be a partition of $n$ ($n=\lambda_1+\lambda_2+\cdots + \lambda_k$ and $\lambda_1\ge \lambda_2\ge\cdots$). There is an ordering on the collection of all the $\lambda$-tabloids: Let $\{...
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Group of rectangular Young Diagrams

Consider set $A$ of all Young diagrams in rectangle $m\times n$. Is there a way to define dot operation $A\times A \to A$, so $(A, \cdot)$ would be a group (preferably Abelian)?
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Reading off tensor index symmetries from a Young Tableau

I'm reading a paper which has given the index symmetries in terms of a Young Tableau which I'm having trouble understanding e.g. one is of the form $[\mu][\nu]$ $[\rho][\sigma]$ I understand that if ...
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Standard Young Tableaux

I'm learning about Young Tableaux.The number of standard Young tableaux of size n can can be generated by the recurrence relation: $a(n)=a(n-1)+(n-1)a(n-2)$ By definition, A standard Young tableau (...
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Tensor product of Schur functors.

I'm trying to prove $\mathbb{S}_{\mu}V\otimes D=\mathbb{S}_{\mu +1^{n}}V$ as representations of GL$_{n}(\mathbb{C})$ and dim$V=n$ as complex vector space, but without using that the characters are ...
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$\eta$-value of a partition and its meaning

The $\eta$-value of an integer partition $\lambda = \big( \lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_k \geq 0 \big)$ is defined as \begin{equation} \eta \big( \lambda \big) \ := \ \sum_{i=1}^k ...
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Young projectors in Fulton and Harris

In Section 4 of Fulton and Harris' book Representation Theory, they give the definition of a Young tableau of shape $\lambda = (\lambda_1,\dots,\lambda_k)$ and then define two subgroups of $S_d$, the ...
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Kostka Numbers for “Partial Contents”?

The Kostka number $K_{\lambda \mu}$ gives the number of SSYT of shape $\lambda$ with content $\mu$. Are there any results regarding Kostka numbers for "partial contents" (I do not know if there is a ...
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Young tablaux and its shape: a basic question

Let $D$ and $D'$ be Young tablaux w.r.t. partitions $(n_1\ge \cdots n_r)$ and $(m_1\ge \cdots \ge m_s)$ of $n$. Suppose that If $\alpha,\beta$ are in same row of $D$ then $\alpha,\beta$ ...
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What is the mistake in finding the irreps of $SU(3)$ multiplets $6 \otimes 8$, or $15 \otimes \bar{15}$?

I wrote a Mathematica paclet that can be used to find irreducible representations of $SU(n)$. Specifically, given a product of $SU(n)$ multiplets, it can compute the corresponding sum. To test the ...
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$SU(N)$ irreps and Young-Diagrams

I have read (source not publicly available) that there is a one-to-one correspondence between young-diagrams with less then $n$ rows and the irreducible representations (irrep) of $SU(n)$. Is this ...
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Young Diagrams, singlets and ignoring the first column

Consider the following Young diagram of $SU(5)$ Now from what I have learned we can ignore the first column of this (since it has 5 elements) and thus write it as: Now if we write down the tensor ...
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Adjoint representation of SU(2) and Young Tableaux

Suppose we have the fundamental representation of $SU(N)$ (represented by a box) which acts on the vector space $V$. Then, the irreducible representation of $SU(N)$ can be found using the rules for ...
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Young Tableaux of $SU(N)$: Group or Algebra?

I am slightly confused about the use of Young Tableaux in the context of the Lie group and Lie algebras of $SU(N)$. A Young Tableaux has an associated representation which acts on a tensor e.g. $\psi_{...
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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 $...
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Representation of $S_n$ that restricts to regular of $S_{n-2}$

I look for a representation $V$ of dimension $n-2!$ (over complex numbers) of the symmetric group $S_n$, such that the restriction to $S_{n-2}$ is the regular representation of $S_{n-2}$. There a ...
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Semi-standard Young Tableau symmetries from RSK on n-letter words

Preamble: Every n-letter word $w$ constructed from a n-letter alphabet is uniquely mapped by the RSK algorithm to a couple of tableaux P and Q, where P is a semi-standard Young tableau (SSYT) and Q is ...
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Counting Semistandard Young Tableaux For Triangular Shapes?

If $k \leq n$ I denote the Young diagram with shape $(n,n-1,n-2,\ldots,1)$ by $\lambda^{n,n-1,\ldots,1}$. I write $f^{\lambda_n^{n,n-1,\ldots,1}}$ to count the number of semistandard Young tableaux ...
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Counting the number of semistandard Young Tableaux with maximum entry $n.$ Reference/Formula request

Question: If $k \leq n$ let $\lambda_k$ be a Young diagram with square $k \times k$ shape. I write $\#_{\lambda_{k}^n}$ to count the number of semistandard Young tableaux with shape $\lambda_k$ and ...
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What is the asymptotic behavior of the number of involutions?

Sequence A000085 in the On-Line Encyclopedia of Integer Sequences counts the number $A_n$ of involutions on $n$ letters, and also, the number of Young tableaux with $n$ cells. I am curious, what is ...
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What is a tableau?

Is a tableau in mathematics the same as a Young tableau? I came across the term in the introduction of this paper, where they say they want to generalise Pascal triangles to a tableau. What do they ...
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Murnaghan-Nakayama rule for general dimension of a hook

Let $m,r\in \mathbb{N}, n=rm$. The character of symmetric group $\chi^{\lambda}$ where $\lambda$ is of the from $(k,1^{n-k})$ evaluated at the conjugacy class of $S_{n}$ of the from $(r,\ldots,r)$, ...
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Combinatorial identity

Let $a_1,\cdots,a_n$ be n positive consecutive integers. So I want to know if there exists a close combinatorial form for the coefficient of $x^k$ in $$(x+a_1)(x+a_2)\ldots (x+a_n) .$$ In ...
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Hook-content polynomial

Let $\lambda$ denote the hook of size $d$ for a fixed integer $d>0$. For $d=2$ there are two kinds of hooks for $d=3$ there are 3 different kind and so on. $c(\Box)$ denote the content of the $\...
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Schur function principal specialisation

Let $s_{\lambda}(p_1,p_2,)$ denote schur function in power-sum symetric basis. More precisely $$ s_{\lambda}=\sum_{\mu}\chi_{\mu}^{\lambda}p_{\mu}|k_{\mu}|.(*)$$ $p_i=\sum_k x_k^i$. For more ...
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Why is the ladder tableau of an $e$-restricted partition $e$-restricted?

Lemma 3.40 on page 46 in Mathas's "Iwahori-Hecke Algebras and the Symmetric Group" states Suppose that $\lambda$ is an $e$-restricted partition of $n$. Then the ladder tableau $\mathfrak{l}_e^\...
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Young operator acting on tensor with symmetries

I am performing an irreducible decomposition of a tensor of rank 4, where it is symmetric in the first two indices: $T_{abmn} = T_{bamn}$. In English notation, the Young tableaux I need to evaluate ...
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Is there a explicit formula for the number of Semi-standard Young Tableaux over $\{1,\dots,n\}$ for a given partition $\lambda$ and a given type $\mu$

I was given an exercise to give all SSYT over $\{1,\dots,12\}$ of shape $\lambda=(4,4,3,1)$ and type $\mu=(4,2,2,2,2,0,\dots,0)$. Now I was wondering if there is an formula to say something about the ...
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Decomposing bimodule into irreducibles

Let $M$ be an $(S_n,S_m)$-bimodule. In particular, $M$ is left $S_n$-module, and a right $S_m$-module, and as such admits decompositions into its irreducible $S_n$-modules, $$M \cong \bigoplus_\...
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265 views

Reference request: Representation theory over fields of characteristic zero

Many representation theory textbooks and online resources work with the field of complex numbers or more generally algebraically closed fields of characteristic zero. I am looking for a good textbook ...
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Polynomials invariant under specific permutations or subgroups of $S_n$.

The Schur polynomials or the power symmetric polynomials are such that they are invariant under the whole $S_n$. Are there polynomials which are invariant only under a chosen permutation group element ...
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Basis of the representations of the B and C series Lie groups

As is well-known, the representations of $SU(n)$ are labelled by Young diagrams. Moreover, there exists a canonical basis of each representation labelled by all the possible tableaux of the diagram. ...
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Why isn't this Young tableaux zero?

I am trying to understand Young tableaux in the context of $SU(2)$ irreducible representations (this is a physics course so we use them just as a tool without much care in the meaning). As I ...
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a Standard Young Tableau symmetry and the non-attacking rook problem.

Any standard or semi-standard Young tableau T of shape $\lambda$ of n can be transformed into an other tableau T´ of $\lambda$ by an operation flip(T) = T´. This operation 'flip' consists of the ...
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Tensoring Young Tableaus

As we all know very well, the finite dimensional irreducible representations of the compact Lie groups $SU(N)$ are labelled by Young tableaus. Now when we tensor two irreducible representations we get ...
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Question about RSK correspondence and semi-standard young tableaux

I am reading the book of "Young tableaux with applications" by Fulton W. The book says that by using row-insersion, we can find: a correspondence between a pair of tableau $(P,Q)$ and a matrix (or ...
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Questions about young tableaux product operation

I am reading the book "Young tableaux with application" by Fulton W. There is claim says that "The product operation makes the set of tableaux into an associative monoid" Where the product ...
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Part sum of a partition of a positive integer $n$

Let $\lambda = (\lambda_1,\lambda_2,\ldots,\lambda_p)$ be a partition of a positive integer $n$. We call $\lambda_i$ a part of $\lambda$. I am interested in the sum of arbitrary parts of $\lambda$. ...
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What's the relation between standard Young tableaux and Catalan number?

From wikipedia, I know some basic facts about Catalan number and Young tableaux. Moreover, I know that Catalan number $C_n$ is the number of triangulations of a $n+2$-gon. What's the relation ...
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What are these numbers associated to partitions?

In doing some "research" I came across the following numbers $c(\lambda)$ associated to a partition $\lambda$. I'm sure these are not new numbers and are probably counting some type of Young tableaux,...