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

31 pages; 8 figures

References in corpus (2)

• Existence of Kirillov-Reshetikhin crystals for nonexceptional types
• On tensor products of polynomial representations

Cited by in corpus (7)

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• Cyclic Sieving and Plethysm Coefficients
• Quantum extremal loop weight modules and monomial crystals
• Kirillov-Reshetikhin crystals $B^{1,s}$ for ${\widehat{\mathfrak{sl}}}_n$ using Nakajima monomials
• Order polynomial product formulas and poset dynamics