Realization of affine type A Kirillov-Reshetikhin crystals via polytopes

TL;DR

This paper constructs an explicit polytope-based realization of affine type A Kirillov-Reshetikhin crystals, demonstrating their structure and properties using combinatorial and crystal basis theories.

Contribution

It introduces a new polytope model for affine type A Kirillov-Reshetikhin crystals and proves their isomorphism with known crystal bases.

Findings

Polytope structure models affine type A crystals

Established isomorphism with Kashiwara's crystal bases

Defined a bijective map satisfying weak promotion properties

Abstract

On the polytope defined in Feigin, Fourier, and Littelmann (2011), associated to any rectangle highest weight, we define a structure of an type $A_n$-crystal. We show, by using the Stembridge axioms, that this crystal is isomorphic to the one obtained from Kashiwara's crystal bases theory. Further we define on this polytope a bijective map and show that this map satisfies the properties of a weak promotion operator. This implies in particular that we provide an explicit realization of Kirillov-Reshetikhin crystals for the affine type $A^{(1)}_n$ via polytopes.