On the uniqueness of promotion operators on tensor products of type A crystals

TL;DR

This paper characterizes the promotion operators on tensor products of type A crystals, showing they exist only for certain highest weight crystals and are unique in specific cases, supporting broader conjectures about affine crystals.

Contribution

It proves that only rectangular highest weight crystals admit promotion operators and establishes the uniqueness of these operators on tensor products of such crystals.

Findings

Promotion operators exist only on rectangular highest weight crystals.

On tensor products of two such crystals, the promotion operator is unique.

Results support Kashiwara's conjecture on the structure of affine crystals.

Abstract

The affine Dynkin diagram of type $A_n^{(1)}$ has a cyclic symmetry. The analogue of this Dynkin diagram automorphism on the level of crystals is called a promotion operator. In this paper we show that the only irreducible type $A_n$ crystals which admit a promotion operator are the highest weight crystals indexed by rectangles. In addition we prove that on the tensor product of two type $A_n$ crystals labeled by rectangles, there is a single connected promotion operator. We conjecture this to be true for an arbitrary number of tensor factors. Our results are in agreement with Kashiwara's conjecture that all `good' affine crystals are tensor products of Kirillov-Reshetikhin crystals.