On the Polytope Model and Near End node Isomorphisms of Type $A$ Kirillov--Reshetikhin Crystals

TL;DR

This paper develops an inductive method to construct paths in the crystal graph of type A Kirillov-Reshetikhin crystals, providing explicit formulas for certain cases and characterizing affine crystal isomorphisms.

Contribution

It introduces an inductive formula for paths in the polytope realization of KR crystals and explicitly describes affine isomorphisms for specific indices.

Findings

Constructed an inductive formula for paths in the crystal graph.

Provided explicit formulas for cases with i ≤ 2 or i ≥ n-1.

Determined the image of elements under affine crystal isomorphisms.

Abstract

We prove an inductive formula to construct a path from the highest weight element to any given vertex in the crystal graph of the polytope realization of the Kirillov-Reshetikhin crystal $KR^{i,m}$ of type $A$. For $i \leq 2$ or $i \geq n-1$, we provide explicit formulas of the same by only using the lowering crystal operators and in those cases, using these paths, we determine the explicit image of any element under the affine crystal isomorphisms between the polytope and the tableau realizations of the Kirillov-Reshetikhin crystals.