David B Rush - Cyclic Sieving and Plethysm Coefficients

dmtcs:2509 - Discrete Mathematics & Theoretical Computer Science, January 1, 2015, DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015) - https://doi.org/10.46298/dmtcs.2509
Cyclic Sieving and Plethysm Coefficients

Authors: David B Rush 1

  • 1 Department of Mathematics [MIT]

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.

Volume: DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015)
Section: Proceedings
Published on: January 1, 2015
Imported on: November 21, 2016
Keywords: plethysms,Schützenberger involution,<i>jeu-de-taquin</i> promotion,canonical bases,Kashiwara crystals,cyclic sieving phenomenon,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]
    Source : OpenAIRE Graph
  • Problems in Complex Analysis; Funder: National Science Foundation; Code: 1006294

Linked publications - datasets - softwares

Source : ScholeXplorer IsRelatedTo ARXIV 0806.3131
Source : ScholeXplorer IsRelatedTo DOI 10.1007/s10801-009-0182-3
Source : ScholeXplorer IsRelatedTo DOI 10.48550/arxiv.0806.3131
  • 0806.3131
  • 10.1007/s10801-009-0182-3
  • 10.1007/s10801-009-0182-3
  • 10.48550/arxiv.0806.3131
On the uniqueness of promotion operators on tensor products of type A crystals

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