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

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.