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3
votes
1answer
44 views

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 ...
6
votes
1answer
258 views

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 ...
8
votes
0answers
155 views

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 ...
10
votes
1answer
522 views

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 ...
4
votes
0answers
101 views

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 ...
6
votes
3answers
330 views

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 ...
1
vote
1answer
167 views

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 ...
12
votes
2answers
464 views

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 ...
13
votes
2answers
823 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 ...
7
votes
0answers
232 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. ...
1
vote
2answers
231 views

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 ...
15
votes
1answer
564 views

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, ...
7
votes
1answer
259 views

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 ...
7
votes
1answer
557 views

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 ...
5
votes
2answers
195 views

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 ...
4
votes
1answer
205 views

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 ...
9
votes
2answers
289 views

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 ...
10
votes
1answer
441 views

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 ...
4
votes
1answer
182 views

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. ...
8
votes
3answers
583 views

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} ...
5
votes
0answers
189 views

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} ...
8
votes
0answers
168 views

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 ...
7
votes
2answers
333 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 ...
14
votes
1answer
513 views

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 ...
13
votes
1answer
414 views

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 ...
4
votes
3answers
527 views

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 ...
2
votes
1answer
187 views

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 ...
10
votes
1answer
460 views

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 ...
7
votes
0answers
278 views

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 ...
5
votes
0answers
156 views

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 $\lambda$ ...
2
votes
1answer
168 views

Dimension of spaces of invariants/tableaux functions

The Hook lenght formula gives the number of standard Young tableaux on a given diagram. A variant gives the number of semistandard tableuax. Does there exist a formula for counting "weighted ...
6
votes
1answer
484 views

Interpreting the Garnir relations in terms of the Yang-Baxter equation

One possible construction of the Specht modules goes as follows. Given a partition $\lambda$ of $n$, we can write down Young's seminormal form for the representation of $S_n$ corresponding to ...
11
votes
2answers
886 views

Sym(V ⊕ ∧² V) isomorphic to direct sum of all Schur functors of V

Let $V$ be a finite-dimensional $K$-vector space. Then, the symmetric power $\mathrm{Sym}\left(V\oplus \wedge^2 V\right)$ is isomorphic to the direct sum of all Schur functors applied to $V$ (each one ...
3
votes
1answer
274 views

Is there an account of the algebra of highest weight tensors?

The context for this question is Schur-Weyl duality. Let $V$ be a vector space. For $r>0$ consider $\otimes^rV$. This has commuting actions of the symmetric group $S(r)$ and $G=GL(V)$. My question ...
4
votes
2answers
802 views

A sum involving irreducible characters of the symmetric group

Recently, during the research, I came across a sum, denoted by $H(n,L)$, involving irreducible characters of the symmetric group, \begin{equation} H(n,L)\colon=\sum_{Y_{i,j,w}} ...
4
votes
1answer
607 views

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} ...
10
votes
4answers
1k 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, ...
6
votes
2answers
684 views

What is the most general “two in one row for A & in one column for B” theorem?

Let $A$ and $B$ be two Young tableaux, i. e. Young diagrams filled with the numbers $1$, $2$, ..., $n$ for some $n$ (not necessarily the same $n$). (They need not be semistandard.) (a) (Etingof's ...
13
votes
3answers
535 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 ...
5
votes
1answer
226 views

Asymptotics of symmetry types of tensors

Introduction Let's fix $m\in \mathbb N$. For each n, the unitary group $\mathbf U(m)$ is represented in the space of tensors of rank $n$ over $\mathbb C^m$ $$V_{n,m}=\bigotimes_{k=1}^n \mathbb C^m$$ ...
12
votes
3answers
1k views

Equivalence relations on permutations and pattern avoidance

I'm working on the interaction between equivalence relations on permutations and pattern avoidance. I've only considered Knuth equivalence and cyclic shifts until now and I'm looking for other ...
20
votes
3answers
724 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 representation ...
7
votes
3answers
621 views

Symmetric Powers, Tableau and Wreath Products

Let V and W be irreducible representations of $S_n$ and $S_m$ over a field of characteristic 0. Then the Littlewood-Richardson coefficients allow us to compute the isomorphism type of the induced ...
15
votes
3answers
893 views

Young's lattice and the Weyl algebra

Let L be the lattice of Young diagrams ordered by inclusion and let Ln denote the nth rank, i.e. the Young diagrams of size n. Say that lambda > mu if lambda covers mu, i.e. mu can be obtained ...