A generalization of the alcove model and its applications

TL;DR

This paper introduces the quantum alcove model, a generalization of the alcove model, to describe tensor products of crystals in affine types, providing explicit isomorphisms and energy function formulas.

Contribution

It extends the alcove model to a quantum version applicable to affine types and offers explicit crystal isomorphisms in types A and C.

Findings

Quantum alcove model describes tensor products of Kirillov-Reshetikhin crystals.

Explicit affine crystal isomorphisms are constructed for types A and C.

An efficient formula for the energy function is provided.

Abstract

The alcove model of the first author and A. Postnikov uniformly describes highest weight crystals of semisimple Lie algebras. We construct a generalization, called the quantum alcove model. In joint work of the first author with S. Naito, D. Sagaki, A. Schilling, and M. Shimozono, this was shown to uniformly describe tensor products of column shape Kirillov-Reshetikhin crystals in all untwisted affine types; moreover, an efficient formula for the corresponding energy function is available. In the second part of this paper, we specialize the quantum alcove model to types $A$ and $C$. We give explicit affine crystal isomorphisms from the specialized quantum alcove model to the corresponding tensor products of column shape Kirillov-Reshetikhin crystals, which are realized in terms of Kashiwara-Nakashima columns.