Jeu de taquin dynamics on infinite Young tableaux and second class particles

Dan Romik; Piotr Śniady

doi:10.1214/13-aop873

A 'Nice' Bijection for a Content Formula for Skew Semistandard Young Tableaux

The Electronic Journal of Combinatorics ◽

10.37236/1635 ◽

2002 ◽

Vol 9 (1) ◽

Cited By ~ 1

Author(s):

Martin Rubey

Keyword(s):

Generating Function ◽

Young Tableaux ◽

Jeu De Taquin ◽

Enumerative Combinatorics ◽

Bijective Proof

Based on Schützenberger's evacuation and a modification of jeu de taquin, we give a bijective proof of an identity connecting the generating function of reverse semistandard Young tableaux with bounded entries with the generating function of all semistandard Young tableaux. This solves Exercise 7.102 b of Richard Stanley's book 'Enumerative Combinatorics 2'.

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Jeu de taquin, uniqueness of rectification and ultradiscrete KP

Journal of Integrable Systems ◽

10.1093/integr/xyz012 ◽

2019 ◽

Vol 4 (1) ◽

Author(s):

Shinsuke Iwao

Keyword(s):

Time Evolution ◽

Integrable System ◽

Combinatorial Problem ◽

Young Tableaux ◽

Kp Equation ◽

Jeu De Taquin ◽

The Relationship

Abstract In this article, we study tropical-theoretic aspects of the ‘rectification algorithm’ on skew Young tableaux. It is shown that the algorithm is interpreted as a time evolution of some tropical integrable system. By using this fact, we construct a new combinatorial map that is essentially equivalent to the rectification algorithm. Some of properties of the rectification can be seen more clearly via this map. For example, the uniqueness of a rectification boils down to an easy combinatorial problem. Our method is mainly based on the two previous researches: the theory of geometric tableaux by Noumi–Yamada, and the study on the relationship between jeu de taquin slides and the ultradiscrete KP equation by Mikami and Katayama–Kakei.

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Minimal Orbits of Promotion

The Electronic Journal of Combinatorics ◽

10.37236/5885 ◽

2017 ◽

Vol 24 (1) ◽

Cited By ~ 1

Author(s):

Kevin Purbhoo ◽

Donguk Rhee

Keyword(s):

Symmetric Group ◽

Young Tableaux ◽

Jeu De Taquin ◽

Rectangular Shape ◽

Standard Young Tableaux

We give a bijection between the symmetric group $S_n$, and the set of standard Young tableaux of rectangular shape $m^n$, $m \geq n$, that have order $n$ under jeu de taquin promotion.

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Cyclic Sieving and Plethysm Coefficients

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2509 ◽

2015 ◽

Vol DMTCS Proceedings, 27th... (Proceedings) ◽

Author(s):

David B Rush

Keyword(s):

Fixed Points ◽

Schur Function ◽

Young Tableaux ◽

Jeu De Taquin ◽

Cyclic Action ◽

Domino Tableaux ◽

Ribbon Tableaux ◽

International Audience ◽

Cyclic Sieving ◽

Element Set

International audience A combinatorial expression for the coefficient of the Schur function $s_{\lambda}$ in the expansion of the plethysm $p_{n/d}^d \circ s_{\mu}$ is given for all $d$ dividing $n$ for the cases in which $n=2$ or $\lambda$ is rectangular. In these cases, the coefficient $\langle p_{n/d}^d \circ s_{\mu}, s_{\lambda} \rangle$ is shown to count, up to sign, the number of fixed points of an $\langle s_{\mu}^n, s_{\lambda} \rangle$-element set under the $d^e$ power of an order $n$ cyclic action. If $n=2$, the action is the Schützenberger involution on semistandard Young tableaux (also known as evacuation), and, if $\lambda$ is rectangular, the action is a certain power of Schützenberger and Shimozono's <i>jeu-de-taquin</i> promotion.This work extends results of Stembridge and Rhoades linking fixed points of the Schützenberger actions to ribbon tableaux enumeration. The conclusion for the case $n=2$ is equivalent to the domino tableaux rule of Carré and Leclerc for discriminating between the symmetric and antisymmetric parts of the square of a Schur function. Une expression combinatoire pour le coefficient de la fonction de Schur $s_{\lambda}$ dans l’expansion du pléthysme $p_{n/d}^d \circ s_{\mu}$ est donné pour tous $d$ que disent $n$, dans les cas où $n=2$, ou $\lambda$ est rectangulaire. Dans ces cas, le coefficient $\langle p_{n/d}^d \circ s_{\mu}, s_{\lambda} \rangle$ se montre à compter, où l’on ignore le signe, le nombre des point fixés d’un ensemble de $\langle s_{\mu}^n, s_{\lambda} \rangle$ éléments sous la puissance $d^e$ d’une action cyclique de l’ordre $n$. Si $n=2$, l’action est l’involution de Schützenberger sur les tableaux semi-standard de Young (aussi connu sous le nom des évacuations), et si $\lambda$ est rectangulaire, l’action est une certaine puissance de l’avancement jeu-de-taquin de Schützenberger et Shimozono.Ce travail étend les résultats de Stembridge et Rhoades, liant les point fixés des actions de Schützenberger aux tableaux de ruban. Pour le cas $n=2$ , la conclusion est équivalent à la règle des tableaux de dominos de Carré et Leclerc, qui distingue entre les parties symétriques et asymétriques du carré d’une fonction de Schur.

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Symmetry properties of the Novelli-Pak-Stoyanovskii algorithm

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2393 ◽

2014 ◽

Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽

Author(s):

Robin Sulzgruber

Keyword(s):

Young Tableau ◽

Sorting Algorithm ◽

Young Tableaux ◽

Hook Length ◽

Jeu De Taquin ◽

Bijective Proof ◽

Symmetry Properties ◽

International Audience ◽

Standard Young Tableaux ◽

Standard Young Tableau

International audience The number of standard Young tableaux of a fixed shape is famously given by the hook-length formula due to Frame, Robinson and Thrall. A bijective proof of Novelli, Pak and Stoyanovskii relies on a sorting algorithm akin to jeu-de-taquin which transforms an arbitrary filling of a partition into a standard Young tableau by exchanging adjacent entries. Recently, Krattenthaler and Müller defined the complexity of this algorithm as the average number of performed exchanges, and Neumann and the author proved it fulfils some nice symmetry properties. In this paper we recall and extend the previous results and provide new bijective proofs.

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Promotion of Increasing Tableaux: Frames and Homomesies

The Electronic Journal of Combinatorics ◽

10.37236/6836 ◽

2017 ◽

Vol 24 (3) ◽

Cited By ~ 3

Author(s):

Oliver Pechenik

Keyword(s):

Schubert Calculus ◽

Young Tableaux ◽

Jeu De Taquin ◽

Average Value ◽

K Promotion ◽

Standard Young Tableaux ◽

Promotion Operator

A key fact about M.-P. Schützenberger's (1972) promotion operator on rectangular standard Young tableaux is that iterating promotion once per entry recovers the original tableau. For tableaux with strictly increasing rows and columns, H. Thomas and A. Yong (2009) introduced a theory of $K$-jeu de taquin with applications to $K$-theoretic Schubert calculus. The author (2014) studied a $K$-promotion operator $\mathcal{P}$ derived from this theory, but observed that this key fact does not generally extend to $K$-promotion of such increasing tableaux. Here, we show that the key fact holds for labels on the boundary of the rectangle. That is, for $T$ a rectangular increasing tableau with entries bounded by $q$, we have $\mathsf{Frame}(\mathcal{P}^q(T)) = \mathsf{Frame}(T)$, where $\mathsf{Frame}(U)$ denotes the restriction of $U$ to its first and last row and column. Using this fact, we obtain a family of homomesy results on the average value of certain statistics over $K$-promotion orbits, extending a $2$-row theorem of J. Bloom, D. Saracino, and the author (2016) to arbitrary rectangular shapes.

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One-dimensional Schubert Problems with Respect to Osculating Flags

Canadian Journal of Mathematics ◽

10.4153/cjm-2015-061-1 ◽

2017 ◽

Vol 69 (1) ◽

pp. 143-185 ◽

Cited By ~ 2

Author(s):

Jake Levinson

Keyword(s):

Moduli Space ◽

Special Interest ◽

Combinatorial Proof ◽

Young Tableaux ◽

Normal Curve ◽

Jeu De Taquin ◽

One Dimensional ◽

First Order ◽

Real Geometry ◽

Rational Normal Curve

AbstractWe consider Schubert problems with respect to flags osculating the rational normal curve. These problems are of special interest when the osculation points are all real. In this case, for zerodimensional Schubert problems, the solutions are “ as real as possible”. Recent work by Speyer has extended the theory to the moduli space allowing the points to collide. This gives rise to smooth covers (ℝ), with structure and monodromy described by Young tableaux and jeu de taquin.In this paper, we give analogous results on one-dimensional Schubert problems over .Their(real) geometry turns out to be described by orbits of Schützenberger promotion and a related operation involving tableau evacuation. Over M 0,r, our results show that the real points of the solution curves are smooth.We also find a new identity involving “first-order” K-theoretic Littlewood-Richardson coefficients, for which there does not appear to be a known combinatorial proof.

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Higher spins from exotic dualisations

Journal of High Energy Physics ◽

10.1007/jhep03(2021)171 ◽

2021 ◽

Vol 2021 (3) ◽

Author(s):

Nicolas Boulanger ◽

Victor Lekeu

Keyword(s):

Infinite Number ◽

Arbitrary Number ◽

Degrees Of Freedom ◽

Young Tableaux ◽

Spacetime Dimension ◽

Massless Field ◽

Free Level ◽

Higher Spins

Abstract At the free level, a given massless field can be described by an infinite number of different potentials related to each other by dualities. In terms of Young tableaux, dualities replace any number of columns of height hi by columns of height D − 2 − hi, where D is the spacetime dimension: in particular, applying this operation to empty columns gives rise to potentials containing an arbitrary number of groups of D − 2 extra antisymmetric indices. Using the method of parent actions, action principles including these potentials, but also extra fields, can be derived from the usual ones. In this paper, we revisit this off-shell duality and clarify the counting of degrees of freedom and the role of the extra fields. Among others, we consider the examples of the double dual graviton in D = 5 and two cases, one topological and one dynamical, of exotic dualities leading to spin three fields in D = 3.

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