Proceedings of the 28-th International Conference on Formal Power Series and Algebraic Combinatorics
4-8 Jul 2016 Vancouver, British Columbia (Canada)
Affine type A geometric crystal structure on the Grassmannian
Gabriel Frieden  1  
1 : Department of Mathematics [Ann Arbor]
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA -  États-Unis

We construct a type A(1) n−1 affine geometric crystal structure on the Grassmannian Gr(k, n). The tropicalization of this structure recovers the combinatorics of crystal operators on semistandard Young tableaux of rectangular shape (with n − k rows), including the affine crystal operator e 0. In particular, the promotion operation on these tableaux essentially corresponds to cyclically shifting the Plu ̈cker coordinates of the Grassmannian. 

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