A uniform realization of the combinatorial $R$-matrix
Abstract
Kirillov-Reshetikhin crystals are colored directed graphs encoding the structure of certain finite-dimensional representations of affine Lie algebras. A tensor products of column shape Kirillov-Reshetikhin 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 -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 Sch\"utzenberger's sliding game (jeu de taquin) for Young tableaux, which realizes the combinatorial -matrix in type . Our construction is in terms of certain combinatorial moves, called quantum Yang-Baxter moves, which are explicitly described by reduction to the rank 2 root systems. We also show that the quantum alcove model does not depend on the choice of a sequence of alcoves joining the fundamental one to a translation of it.
Cite
@article{arxiv.1503.01765,
title = {A uniform realization of the combinatorial $R$-matrix},
author = {Cristian Lenart and Arthur Lubovsky},
journal= {arXiv preprint arXiv:1503.01765},
year = {2015}
}
Comments
arXiv admin note: text overlap with arXiv:1112.2216