Questions tagged [young-tableaux]

For standard Young tableaux, semistandard Young tableaux, and other related two-dimensional arrays of numbers like plane partitions. Including their combinatorial theory and their application in representation theory and algebraic geometry.

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A criterion for when a symmetrized decomposable tensor is nonzero

In the problem I am interested in, one has a collection of vector spaces, all $2$-dimensional, denoted by $V_{i, j}$, where $1 \leq i \neq j \leq n$. In other words, the set of indices is $$S = \{(i, ...
11 votes
1 answer
626 views

What are the homological properties of Young's lattice?

Young's lattice $Y$ is a graded poset and a distributive lattice whose elements are all the partitions of $n$ for $n \in \mathbb{N}$ with the poset relation coming from inclusion of Young diagrams. ...
28 votes
3 answers
1k views

Inequality for hook numbers in Young diagrams

Consider a Young diagram $\lambda = (\lambda_1,\ldots,\lambda_\ell)$. For a square $(i,j) \in \lambda$, define hook numbers $h_{ij} = \lambda_i + \lambda_j' -i - j +1$ and complementary hook numbers $...
2 votes
1 answer
110 views

A problem about the existence of increasing coloring groups

Got stuck on this one for months. Given a sequence of non decreasing positive integers $a_1, .., a_n$, let there be $a_i$ balls labeled the number i, for each $1 \leq i \leq n$. Suppose there is a k ...
11 votes
1 answer
791 views

Dual of a Specht module

For a partition $\mu$ of $n$, let $S^{\mu}$ be the associated Specht module, defined over $\mathbb{Z}$. For any field $k$, we can tensor $S^{\mu}$ with $k$ to get a representation $S^{\mu}_k$ of the ...
4 votes
0 answers
143 views

Major indices of standard tableaux of shapes obtained from addable cells of a given Young diagram

I have a "very" indirect proof that the following fact is true for every Young diagram $\lambda \vdash n$ and every $r \in \{0,\dotsc,n\}$: \begin{equation} d_\lambda = \sum_{a \in \mathrm{...
5 votes
1 answer
210 views

Schur functors = Weyl functors in characteristic zero?

I asked this question on Math Stack Exchange https://math.stackexchange.com/questions/4789924/schur-functors-weyl-functors-in-characteristic-zero, but I got no answers, so I ask the same question here....
36 votes
1 answer
1k views

Errata for Fulton's "Young tableaux"

Fulton's Young tableaux is one of the best texts on the subject, one which I often recommend and cite for reference. Unlike Fulton/Lang and Fulton/Harris, it is neither an early-dawn draft nor a ...
1 vote
1 answer
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Combinatorics behind certain induction of characters of the Coxeter group of type $B_n$

Let $W_n$ be a Coxeter group of type $B_n$ with $n\geq 1$. Concretely, it is generated by a set of simple reflexions $S = \{s_1,\ldots ,s_n\}$ which satisfy the relations $s_i^2 = 1, s_is_j=s_js_i$ as ...
13 votes
1 answer
674 views

Most computationally efficient Littlewood-Richardson rule

There are many, many different versions of the Littlewood-Richardson rule: the original characterization via Yamanouchi words, Remmel's version, a description via the Poirier-Reutenauer bialgebra, the ...
8 votes
2 answers
714 views

Bender-Knuth involutions for symplectic (King) tableaux

First let me recall the combinatorial theory of the characters of $\mathfrak{gl}_m$, a.k.a., Schur polynomials. For a partition $\lambda$, a semistandard Young tableaux of shape $\lambda$ is a filling ...
5 votes
1 answer
110 views

geometric meaning to pairs of SYT indexing for the basis of cohomology ring of full flag variety

For Grassmannians, the Schubert cells can be indexed by certain Young Tableaux, whose partition determines the dimensions of intersections of the chosen subspace with the standard complete flag. For ...
2 votes
0 answers
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Skewed plane partition with only row fillings reversed

The number of plane partitions in a bounded box is well-studied and dates back to MacMahon, at the start of this paper by Sam Hopkins and Tri Lai, p9, they summarized current results on the ...
-1 votes
1 answer
163 views

Orthogonality of irreducible and non-isomorphic representations [closed]

Let V and W be any two subspaces of $(\mathbb{C}^d)^{\otimes n}$ such that there exists two irreducible and non-isomorphic representations $\rho_V: G \to GL(V)$ and $\rho_W: G \to GL(W)$. Does this ...
7 votes
2 answers
265 views

Decomposition of tensors into symmetry classes according to Schur functors

I am mainly looking for references on this subject, as I was unable to find any, at least any that answers what I am looking for to a satisfactory degree. As it is well-known and extremely easy to ...
1 vote
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120 views

Counting certain kinds of Semistandard Young Tableaux

We have a project in which it is natural to count the number of Young Tableau in which part of the weight has been specified. Does anybody know if this idea already appears in the literature? More ...
3 votes
1 answer
250 views

Decomposition of tensor powers of the vector representation of $\frak{sl}_n$

Let $V(\pi_1)$ be the usual vector/matrix representation of the Lie algebra $\frak{sl}_n$, for $n > 2$. A basic fact is the tensor product $V(\pi_1) \otimes V(\pi_1)$ decomposes as $$ V(\pi_1) \...
3 votes
0 answers
107 views

Generalized Gaussian binomial and symmetric chain decomposition

Background Let $\mu = (\mu_1, \ldots, \mu_k)$ be a partition, meaning that $\mu_1 \geq \ldots \mu_k \geq 1$. The Young diagram associated to $\mu$ is given by the set $(r,c) \in \mathbb{N} \times \...
1 vote
1 answer
219 views

hook length formula for plane partitions

The hook length formula give a simple product expression for the number of standard Young tableaux of a given shape $\lambda$, where $\lambda$ is an integer partition, or equivalently, the number of ...
8 votes
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Alternate proof in Fulton–Harris of representation theoretic version of Littlewood–Richardson rule

$\DeclareMathOperator\Ind{Ind}$Let $d = d_1 + d_2$ with $d_1$, $d_2$ positive integers. Let $\lambda$ be a partition of $d_1$ and $\mu$ a partition of $d_2$, so that the Young symmetrizer construction ...
10 votes
2 answers
796 views

Viennot-type geometric description for dual RSK correspondence?

Is a geometric construction of the dual RSK correspondence along the lines of Viennot's "light and shadows construction" written up somewhere? This is a bijective correspondence between 0-1 matrices ...
15 votes
3 answers
843 views

What bijection on permutations corresponds under RS to transpose?

The Robinson-Schensted correspondence is a bijection between elements of the symmetric group and ordered pairs of standard tableaux of the same shape. Some simple operations on tableaux correspond to ...
7 votes
0 answers
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Mistakes in Logan and Shepp's famous paper on Young Tableaux?

In their landmark paper from 1977 named "A Variational Problem for Random Young Tableaux" Logan and Shepp obtained a number of results concerning asymptotic properties of Young Diagrams ...
3 votes
1 answer
504 views

References for applications of Young diagrams/tableaux to Quantum Mechanics

I am interested in knowing more about applications of Young diagrams and Young tableaux to Quantum Mechanics. A friend of mine suggested as a reference the following book: Wybourne, B.G.; "...
51 votes
2 answers
18k views

Is there winning strategy in Tetris ? What if Young diagrams are falling?

Question 1 Is there a winning strategy (algorithm to play infinitely) in Tetris, or is there a sequence of bricks which is impossible to pack without holes? Consider generalized Tetris with Young ...
22 votes
1 answer
957 views

What is the generating function for skew Young diagrams?

The problem This strikes me as a very natural problem which should have been asked (and solved?) already. For each positive integer k, find a nice expression for the following generating function in ...
23 votes
5 answers
2k views

Is there a short proof that the Kostka number $K_{\lambda \mu}$ is non-zero whenever $\lambda$ dominates $\mu$?

This is maybe a little basic for MathOverflow, but I'm hoping it will get some interesting answers. Let $\unrhd$ be the dominance order on partitions of $n \in \mathbb{N}$. For partitions $\lambda$ ...
0 votes
0 answers
166 views

Young tableaux — irreps correspondence for simple complex Lie algebras

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SL{SL}$I have learned that Young tableaux which were originally introduced to study the irreducible representations of finite symmetric groups $S_n$ ...
6 votes
0 answers
110 views

Bijection between forests and skew SYT + Cyclic sieving

Consider the two-row skew shape $\lambda_n = (2n+1,n)/(1)$. The number of standard Young tableaux of this shape is $\binom{3n}{n}-\binom{3n}{n-2}$ (since one can easily biject this to the set of non-...
7 votes
1 answer
162 views

A formula for the generating function of Hoggatt binomials or of some Young tableaux

Let ${\left\langle\matrix {n \cr k}\right\rangle}_r$ denote the $r-$Hoggatt binomials defined by $${{\left\langle\matrix {n \cr k}\right\rangle}_r=\frac{\langle n \rangle_r!}{\langle k \rangle_r! \...
4 votes
2 answers
304 views

LGV scheme for lattice paths that move in non-unit spatial positive steps

In the Lindström–Gessel–Viennot lemma (LGV) applied to the $Z^2$-lattice paths are taken to move in unit spatial-steps in unit time (see here). What do we mean by "time"? In the language of ...
7 votes
1 answer
260 views

Robinson-Schensted-Knuth (RSK) under restriction

I am curious about the following result concerning the Robinson-Schensted insertion procedure. I can formulate a proof via the Schützenberger evacuation operator, but I have struggled to find such an ...
21 votes
2 answers
2k views

Has Reifegerste's Theorem on RSK and Knuth relations received a slick proof by now?

For the notations I am using, I refer to the Appendix at the end of this post. Here is what, for the sake of this post, I consider to be Reifegerste's theorem: Theorem 1. Let $n\in\mathbb N$ and $i\...
3 votes
0 answers
135 views

Counting integer partitions below some Young diagram

Question: Given positive, coprime integers $m<n$, consider the Young diagram $Y$ formed by the lattice points in the Cartesian plane lying below the line from $(0,0)$ to $(m,n)$ and within the ...
5 votes
1 answer
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some confusion about the explicit construction of irreducible representations of $S_n$

In this book chapter, the irreducible representations of the symmetric group $S_n$ is given in terms of polytabloids of a Ferrer's diagram $\lambda$, defined as $e_t = \sum_{\pi \in C_t} \text{sgn}(\...
12 votes
2 answers
359 views

Lattice structure (wrt dominance order) on the set of Young diagrams appearing in the decompositions given by the Littlewood-Richardson rule

The irreducible decomposition of the tensor product of two irreducible representations of GL(n) is described by the Littlewood-Richardson rule. This same rule also governs the decomposition of the ...
5 votes
1 answer
257 views

On a proof involving Young symmetrizers acting on tensor spaces

I hope this is not too elementary for this site, but I already asked something similar on MSE which has not received any attention whatsoever. I am extremely unfamiliar with the algebraic/...
16 votes
4 answers
4k views

Making the branching rule for the symmetric group concrete

This question concerns the characteristic $0$ representation theory of the symmetric group $S_n$. I'm a topologist, not a representation theorist, so I apologize if I state it in an odd way. First, a ...
1 vote
0 answers
126 views

$\mathfrak{sl}_2$-action on Young diagrams

Let $\mathcal{Y}$ be a vector $\mathbb{Q}$-space of all Young diagrams. Denote by $\delta_\lambda$ the Young diagram of the partition $\lambda$ and $c(\square)$ be the content of the square $\...
3 votes
0 answers
149 views

The Grassmann twist-map, an associated semi-group action, and RSK

Let me begin by setting some notation: Let $\mathrm{Mat}_{k,n}(\Bbb{R})$ denote the vector space of all $k \times n$ real-valued matrices. Given $g \in \mathrm{Mat}_{k,n}(\Bbb{R})$ and two (ordered) ...
25 votes
3 answers
1k views

Statistics of irreps of S_n that can be read off the Young diagram, and consequences of Kerov–Vershik

Alexei Oblomkov recently told me about the beautiful theorem of Kerov and Vershik, which says that "almost all Young diagrams look the same." More precisely: take a random irreducible ...
9 votes
0 answers
229 views

Hives for other root systems? [duplicate]

Littlewood-Richardson coefficients $c^\lambda_{\mu\nu}$ have numerous combinatorial interpretations, including the hive model by Knutson and Tao, see here. For other root systems, there are also ...
4 votes
0 answers
206 views

Relationship between $\mathbb{S}^{\nu}V \otimes \mathbb{S}^{\lambda}(V^{*})$ and $\mathbb{S}^{\nu / \lambda}V$

For partition $\mu$ let $\mathbb{S}^{\mu}V = V^{\otimes \mu} \cdot c_{\mu}$, where $c_{\mu}$ is the Young symmetrizer. I'm trying to prove that $\mathbb{S}^{\nu / \lambda}V$ is the polynomial part of $...
0 votes
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Addition theorem for Schur function in multivariable

Working with the following problem Expansion in Schur function of negative binomial exponent I need to find the expansion of $$ s_{\lambda}(x_1 + y , x_2 +y, \ldots, x_n +y)$$ in terms of schur ...
9 votes
0 answers
480 views

Two majs for standard Young tableaux?

Let $\lambda$ be a partition of $n$, and consider its set of standard Young tableaux (SYTs): bijective fillings of the Young diagram of $\lambda$, written in English notation, with the numbers $1$ ...
2 votes
0 answers
115 views

Yamanouchi ribbon tableaux?

Let $s_{\lambda}$ be a Schur function. The set of all such functions are known to be a linear basis of the algebra of symmetric functions. The Littlewood-Richardson coefficients $c_{\nu\mu}^{\lambda}$ ...
1 vote
0 answers
129 views

Status of conjecture of Conrey and Gonek, combinatorial meaning

I was looking at the OEIS on the number of square Young Tableaux. In it Michael Somos referenced a paper of Conrey and Gonek, High Moment's of the Riemann Zeta-Function. Is there an combinatorial ...
1 vote
0 answers
71 views

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 $\...
21 votes
2 answers
1k views

3D generalizations of permutations, RSK correspondence, contingency tables, etc.

I want to gather facts and questions related to 3D generalizations of permutations, RSK correspondence, contingency tables, etc. One reason I am interested in this is because it is potentially related ...
1 vote
1 answer
176 views

Number of paths to a specific vertex in the Young's lattice

Consider the Young's lattice. What is the number of paths starting from the origin (0) to a specific Young diagram? For instance, the Young diagram corresponding to the integer partition 1+1+1 has 1 ...