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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42 views

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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1answer
35 views

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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26 views

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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1answer
23 views

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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2answers
80 views

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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40 views

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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12 views

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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1answer
65 views

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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1answer
22 views

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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20 views

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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27 views

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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1answer
34 views

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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76 views

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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36 views

Alternative definition for up-down border strip tableau definition?

A border strip tableau (sometimes called ribbon tableau, or rim hook tableau) of shape $\lambda$ and type $\mu = (\mu_{1},\mu_{2},...)$ is defined by the following properties: every row and column is ...
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33 views

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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15 views

Is the tenor product symmetric for Young tableau?

Let (a,b) be a Young diagram with $a$ boxes in the first row, and $b$ boxes in the second row. For example, can the following be done to make the calcultion easier? (4,4) $\otimes$ (2,0) $\...
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49 views

Young tableau sanity check

Young diagram of shape $(a,b)$ has $a$ boxes in the 1st row, $b$ boxes in the second row. Objective: decompose the following direct product of irreps, and then determine their dimensions given $\...
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14 views

I need help with the Proof of the bijection between Young tableaux and matrix via Hillman–Grassl

I don’t know much about combinatorics, I’m just getting started on this. I need help with the proof of this (Hillman–Grassl map Φ) The map Φ : RPP(λ) → A(λ) is a bijections If you know of articles ...
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20 views

Obtaining Skew diagram from two Young diagrams

Consider following Young diagrams and Tableaux: The Young diagram of partition $\mu$ is contained in Young diagram of $\lambda$; but not in unique way (I think): (i) We can choose boxes numbered by $...
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3answers
79 views

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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29 views

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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1answer
222 views

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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40 views

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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1answer
59 views

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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1answer
162 views

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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1answer
78 views

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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39 views

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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43 views

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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105 views

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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1answer
101 views

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
89 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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1answer
44 views

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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1answer
218 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
35 views

$\mu$, $\nu$ 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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2answers
160 views

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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129 views

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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46 views

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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33 views

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 $\{t_1\...
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83 views

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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1answer
243 views

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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2answers
232 views

$\eta$-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 ...
5
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1answer
230 views

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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1answer
123 views

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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44 views

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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1answer
161 views

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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62 views

$SU(N)$ 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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1answer
25 views

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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1answer
257 views

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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1answer
353 views

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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86 views

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 $...