A uniform realization of the combinatorial $R$-matrix

TL;DR

This paper provides a uniform combinatorial realization of the affine crystal isomorphism, known as the $R$-matrix, for all untwisted affine Lie types using the quantum alcove model and quantum Yang-Baxter moves.

Contribution

It extends the quantum alcove model to realize the combinatorial $R$-matrix uniformly across all untwisted affine types, generalizing jeu de taquin to all Lie types.

Findings

The quantum alcove model's independence from the alcove sequence.

Explicit description of quantum Yang-Baxter moves for all rank 2 root systems.

A uniform combinatorial framework for the affine crystal isomorphism.

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 $R$-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 Sch\"utzenberger's sliding game (jeu de taquin) for Young tableaux, which realizes the combinatorial $R$-matrix in type $A$. 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.