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

  • Author(s): Bandlow, Jason
  • Schilling, Anne
  • Thiery, Nicolas M.
  • et al.

Published Web Location

https://arxiv.org/pdf/0806.3131.pdf
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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.

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