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

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

2008-06-19

arxiv.org/abs/0806.3131v2

J. Algebraic Combinatorics 31 (2010) 217-251

Mathematics

Combinatorics

31 pages; 8 figures

Scientific paper

10.1007/s10801-009-0182-3

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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