Affine type A geometric crystal structure on the Grassmannian

Gabriel Frieden (1)

1 Department of Mathematics [Ann Arbor]

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.

Domaines: Combinatoire [math.CO]

Gabriel Frieden. Affine type A geometric crystal structure on the Grassmannian. 28-th International Conference on Formal Power Series and Algebraic Combinatorics, Simon Fraser University, Jul 2016, Vancouver, Canada. ⟨10.46298/dmtcs.6393⟩. ⟨hal-02173374⟩