The young-tableaux tag has no usage guidance.

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### Reference Request: Irreducibles of the regular representation of the permutation group is absolutely irreducible

I am writing a paper(physics) where I am using the fact that the irreducible's of the regular representations of the permutation group are absolutely irreducible in the following sense.
If $V$ is an ...

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### Partial orders on tabloids

Let $n \in \mathbb{N}$ and let $\lambda \vdash n$, a partition of $n$. By a $\lambda$-tabloid I mean a row-tabloid of shape $\lambda$. There is a well-known order on the set of $\lambda$-tabloids, ...

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### Relations among Young symmetrizers of non-standard tableaux

For any Young tableau, one can form the Young symmetrizer. I'm naturally interested in young symmetrizers coming from standard tableaux, but I'm forced to look at Young symmetrizers of non-standard ...

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### 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 ...

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### Measures on Young tableaux

I have seen that on the set of Young tableau the Plancherel measure was quite natural to define. I was wondering if other measures were also studied. In particular, a simple exemple which comes to my ...

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### Character sums over a fixed subset of skew tableaux

Let $f(\lambda)$ count the number standard young tableaux of shape $\lambda\vdash n$ and $\lambda=(\lambda_1,\cdots,\lambda_r)$. Let $\mu \vdash k$ be a partition for $k<n$. It is a consequence of ...

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### No limit shape for random Young diagrams under z-measure?

In their paper Random partitions and the Gamma kernel (Advances in Mathematics 194 (2005) 141–202), Borodin and Olshanski state that:
An important difference between the Plancherel measures and ...

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### Uniform generation of Symmetric Plane Partitons

In the conclusion of An Involution Principle-Free Proof of Stanley's Hook-Content Formula Krattenthaler notes that the techniques of the paper might be useful for finding bijective proofs of the ...

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### Young Tableau Box Correlations

Let $T$ be a uniformly random Standard Young Tableau (SYT) of shape $\lambda=(\lambda_1,\cdots,\lambda_k)$ with $|\lambda|=n$. Let $T_{ij}$ denote the value in box $(i,j)$. I'm interested in what can ...

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### A vector version of the Segre embedding: what is the kernel of the ring map?

TL;DR version.
Given a commutative ring $\mathbf{k}$ and $n+m$ "generic" vectors $\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_n, \mathbf{y}_1, \mathbf{y}_2, \ldots, \mathbf{y}_m$ in $\mathbf{k}^k$ ...

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### Is there any good survey on the hook length formula and related topics?

I am recently doing some research related to the hook length formula.
The hook formula counts the number of Young tableaux of certain type.
I find there are plenty of research already been done and ...

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### Nekrasov-Okounkov hook length formula

I am now reading the paper An explicit expansion formula for the powers of the Euler Product in terms of partition hook lengths by Guo-Niu Han. The author rediscovered what he calls Nekrasov-Okounkov ...

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### What's the analogue of a Young symmetrizer in the Brauer algebra?

According to Schur--Weyl duality, the centralizer of $\mathrm{GL}(V)$ acting diagonally on $V^{\otimes N}$ is the group algebra of the symmetric group $\mathbb S_N$. An equivalent formulation is the ...

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### Staircase Schur functions squared

Let $\Delta_n$ be the staircase-shaped partition $(n-1,n-2,\dots,1)$. Are there any non-obvious combinatorial objects that index $s_{\Delta_n}^2$? Here, $s_\lambda$ is the Schur function indexed by ...

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### Frequency of a representation of SO(3)

When generalizing the basic tenets of Fourier Theory to the symmetric group $S_n$, we can define a notion of the frequency of a basis function (i.e. an irreducible representation of $S_n$). In ...

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### counting how many boxes from a given Young tableau contribute to hook length made out of two YTs

Think of a Young diagram as a collection of rows with numbers of elements
$\mu_1 \geq \mu_2 \geq \cdots \geq \mu_d \geq \mu_{d+1}=0$ (and $\mu_k=0$ for $k>d$) and define for $s=(i,j)$ (where $i$ ...

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### Is $LIS(\pi)+LIS(\sigma)+LIS(\sigma\pi^{-1})$ lower bounded?

In the title, $LIS$ stands for the length of longest increasing subsequence and Greek letters stand for permutations from symmetric group $S_n$.
Considering some cases such as ...

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### Transitivity for Schutzenberger involutions on standard Young tableaux

Let $\lambda$ be a partition of $ n$. Let $ SYT(\lambda) $ denote the set of standard Young tableaux of shape $ \lambda $.
For $ i = 1, \dots, n $, let me define permutations $ S_i $ of the set $ ...

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### In general, are 'Young symmetrisers' given by Littlewood-Richardson 'Orthogonal projection Operators'?

Consider $V^{\otimes n}$ where $V$ is vector space and the representation of GL(V) acting in the usual way. Now if I consider tensor products or plethysms of irreducible spaces, this is not in general ...

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### Counting a Modified Class of Standard Young Tableau

Let $\lambda=(\lambda_1,\cdots,\lambda_n)$ be a partition, with $|\lambda|:=N$. Attach an extra box to $\lambda$ to the right end of the $r$'th row. In coordinate form, the last box on row $r$ has ...

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### A variation on Bulgarian solitare

It appears that a variation on Bulgarian solitare has a fixed point regardless
of the starting $n$.
For example, let $n=69$, and consider this partition:
$$
(8,8,7,7,5,5,5,5,5,4,3,3,2,2)
$$
In ...

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### Sum over growing Young tableaux

Let $\lambda_0,\lambda_1,\lambda_2,\lambda_3,\ldots$ be a sequence of Young diagrams, such that each successive diagram is obtained from the prior by the addition of one box (don't forget that the row ...

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### Number of standard Young tableaux with fixed corner entry

For a partition $\lambda=(\lambda_1\geq \lambda_2\geq\ldots\geq \lambda_k)$ of $n$, let the set of standard Young tableau of shape $\lambda$ be denoted by $SYT(\lambda)$ with boxes at $(i,j)$ denoted ...

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### Generating random weak k-bounded reverse plane partitions

Fix a partition $\lambda$. A weak reverse plane partition of shape $\lambda$ is a filling $0\leq \pi_{ij}$ of $\lambda$ with $\pi_{ij}\leq \pi_{kl}$ whenever $i\leq k$ or $j\leq l$. Note that ...

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### Representations of S_n induced from centralizers of elements

Does anyone have a reference for a good description of representations of $S_{n}$ obtained by inducing up from $C_{S_{n}}(\pi)$, for some element $\pi$ of $S_{n}$? (I'd prefer an efficient ...

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### Running the Greene-Nijenhuis Algorithm Backwards

This question is crossposted from math.stackexchange.com, where it remains unanswered.
Let $Y$ be a Young tableau of shape $\lambda:=(\lambda_1,\ldots,\lambda_n)$, where ...

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### calculating Littlewood-Richardson coefficients

It is known that if $\alpha,\beta,\gamma$ are three partitions then the Littlewood-Richardson coefficient $c_{\alpha \beta}^{\gamma}$ is positive when the triple ($\alpha,\beta,\gamma$
) occurs as ...

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### 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 ...

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### 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. ...

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### Flips on standard Young tableaux and descent sets

Consider $T$ to be a standard Young tableau of shape $\lambda$ (in English notation). The descent set of $T$, $Des(T)$, is defined to the set of all positive integers $i$ such that $i+1$ lies strictly ...

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### Conjectural identities for Young symmetrizers and Young-Jucys-Murphy elements

The following questions I have found in my own notes from about 3 years ago. Unfortunately, I lost much of the context; I believe I made these conjectures reading Okounkov-Vershik, arXiv:0503040v3, ...

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### Does this lattice have a name (and literature)?

The "lattice" in the title appears to be a lattice. At least it's a poset, which I define now.
Fix a partition $\lambda$ of $n$ and consider the set of all standard Young tableaux (each of ...

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### On Applications of Murnaghan Nakayama Rule

This question is crossposted at math.stackexchange here and may be beyond the usual scope of the site. The question is located below. In short, I am looking for an accessible explanation of the ...

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### Random RSK and Plancherel Measure

Let $(X_1,X_2,\ldots)$ be a sequence of i.i.d. random variables. It is known that if these random variables are distributed uniformly on the unit interval, then applying the RSK algorithm to this ...

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### Expression of basis vectors of permutation modules in different bases.

This is a cross-post from math.se, because I did not get any answer there:
Write $[n]:=\{1,\ldots,n\}$. For a partition $\lambda\vdash n$, I will write $[\lambda]$ for the Specht module that ...

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### How exactly does Schützenberger promotion relate to Striker-Williams promotion?

Schützenberger promotion, studied (for example) in Richard Stanley, Promotion and Evacuation, 2009, is a permutation of the set of all linear extensions of a finite poset. Since one can identify the ...

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### Generating Random Young Tableaux: A peculiar probability identity

In the paper by Greene, Nijenhuis and Wilf, an algorithm is proposed for generating uniformly random Young tableaux of shape $\lambda$. The algorithm is to uniformly randomly pick a starting cell, and ...

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### Representations of Sym(n) and SL_d

Irreducible representations of the symmetric group Sym$(n)$, and degree-$n$ algebraic representations of SL$_d(\mathbb C)$ for $d\ge n$, can both be classified by Young diagrams with $n$ boxes.
...

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### bijection between number of partitions of 2n satisfying certain conditions with number of partitions of n

Suppose $\lambda = (\lambda_1,\lambda_2,.....,\lambda_k)$ is a partition of $2n$ where $n \in \mathbb N$ satisfying the following conditions:
(1) $\lambda_{k} = 1$.
(2) $\lambda_{i} - \lambda_{i+1} ...

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### Explicit description of isomorphism when decomposing into irreps

I had a question which is slightly more general than this one on mathoverflow: I am looking for an explicit description of the isomorphism
$\mathbb S_\nu(V\otimes W) \cong \bigoplus C_{\lambda\mu\nu} ...

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### An inequality for the ratio of standard Young tableau with {1,2,…,k} in the first row

For a partition $\lambda \vdash n$, define $\dim \lambda$ to be the number of standard Young tableaux of shape $\lambda$, and $\dim \lambda/(k)$ as the number of standard Young tableaux with ...

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### 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 ...

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### Bruhat order and the Robinson-Schensted correspondence

The Robinson-Schensted correspondence is a bijection between elements of the symmetric group $S_n$ and pairs of standard tableaux of the same shape. The symmetric group is partially ordered by the ...

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### Categorifying the equality of product and coproduct of symmetric functions

Littlewood-Richardson coefficients are both multiplicities of $GL_n$ tensor products, and of restrictions of $GL_{m+n}$ representations to $GL_m \times GL_n$. I want to turn this equality of numbers ...

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### RS to RSK correspondence

The RS correspondence is a correspondence which associates to each permutation a pair of standard Young tableaux of the same shape.
The RSK correspondence associates to each integer matrix (with ...

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### Box-dual of a partition - what is it called?

Fix natural numbers $n,m\in\mathbb{N}$. Given a partition $\lambda\vdash d$ with at most $n$ rows (and at most $m$ columns), we can define a partition ...

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### Analogues of the Knuth and Forgotten equivalences on permutations: have they been studied?

Consider a totally ordered alphabet $A$ of $n$ letters. Let $W$ be the set of all words over $A$ which have no two letters equal. Then, for example, we can define the Knuth equivalence on $W$ as the ...

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### The number of rows in a tableau generated by the RSK algorithm.

It is well known that the number of rows in the semistandard Young tableaux correspondent to a two-line array via RSK is equal to the length of the longest (strictly) decreasing subsequence in the ...

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### distribution of Young diagrams

Consider $\Lambda^p(C^n\otimes C^n)=\oplus_{\pi}S_{\pi}C^n\otimes S_{\pi'}C^n$ as
a $GL_n\times GL_n$-module. This space has dimension $\binom {n^2}p$. I would
like any information on the shapes of ...

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### Bound on coefficients in Young tableuax

The following may be known, but I didn't find anything in the literature.
Background:
The irreducible representations of $S_n$ correspond to shapes of Young tableaux with $n$ elements. Let $\lambda$ ...