Journal of Algebraic Combinatorics

, Volume 31, Issue 2, pp 217–251 | Cite as

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

  • Jason Bandlow
  • Anne SchillingEmail author
  • Nicolas M. Thiéry
Open Access


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.


Affine crystal bases Promotion operator Schur polynomial factorization 


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Copyright information

© The Author(s) 2009

Authors and Affiliations

  • Jason Bandlow
    • 1
  • Anne Schilling
    • 2
    Email author
  • Nicolas M. Thiéry
    • 3
    • 4
  1. 1.Department of MathematicsUniversity of PennsylvaniaPhiladelphiaUSA
  2. 2.Department of MathematicsUniversity of CaliforniaDavisUSA
  3. 3.Univ Paris-SudLaboratoire de Mathématiques d’OrsayOrsayFrance
  4. 4.CNRSOrsayFrance

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