On Combinatorial Models for Affine Crystals

TL;DR

This paper establishes a bijection between two combinatorial models for tensor products of Kirillov-Reshetikhin crystals in classical types, enabling easier calculations of key functions in crystal theory.

Contribution

It introduces a bijection between the quantum alcove model and the tableau model for classical affine crystals, simplifying computational approaches.

Findings

Bijection between quantum alcove and tableau models for classical types.

Translation of energy function and R-matrix calculations between models.

Simplification of combinatorial calculations in crystal theory.

Abstract

We biject two combinatorial models for tensor products of (single-column) Kirillov-Reshetikhin crystals of any classical type $A-D$: the quantum alcove model and the tableau model. This allows us to translate calculations in the former model (of the energy function, the combinatorial $R$-matrix, etc.) to the latter, which is simpler.