Cristian Lenart ; Arthur Lubovsky - A uniform realization of the combinatorial $R$-matrix

dmtcs:2491 - 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.2491
A uniform realization of the combinatorial $R$-matrix

Authors: Cristian Lenart 1; Arthur Lubovsky 1

  • 1 Department of Mathematics and Statistics [Albany-USA]

Kirillov-Reshetikhin (KR) crystals are colored directed graphs encoding the structure of certain finite-dimensional representations of affine Lie algebras. A tensor product of column shape KR crystals has recently been realized in a uniform way, for all untwisted affine types, in terms of the quantum alcove model. We enhance this model by using it to give a uniform realization of the combinatorial $R$-matrix, i.e., the unique affine crystal isomorphism permuting factors in a tensor product of KR crystals. In other words, we are generalizing to all Lie types Schützenberger’s sliding game (jeu de taquin) for Young tableaux, which realizes the combinatorial $R$-matrix in type $A$. We also show that the quantum alcove model does not depend on the choice of a sequence of alcoves


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: Kirillov-Reshetikhin crystals,energy function,combinatorial $R$-matrix,quantum alcove model,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]
Funding:
    Source : OpenAIRE Graph
  • Combinatorics of Crystals, Macdonald Polynomials, and Schubert Calculus; Funder: National Science Foundation; Code: 1101264
  • Representation Theory and Schubert Calculus: Combinatorics and Interactions; Funder: National Science Foundation; Code: 1362627

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Source : ScholeXplorer IsRelatedTo DOI 10.1006/jabr.2001.8905
  • 10.1006/jabr.2001.8905
Schensted-Type Correspondence, Plactic Monoid, and Jeu de Taquin for Type Cn

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