Affine type A geometric crystal on the Grassmannian

Abstract

We construct a type An−1(1) geometric crystal on the variety Gr(k,n)×C×, and show that it tropicalizes to the disjoint union of the Kirillov-Reshetikhin crystals corresponding to rectangular tableaux with n−k rows. A key ingredient in our construction is the Z/nZ symmetry on the Grassmannian coming from cyclically shifting the basis of the underlying vector space. We show that a twisted version of this symmetry tropicalizes to combinatorial promotion. Additionally, we use the loop group GLn(C(λ)) to define a unipotent crystal which induces our geometric crystal. We use this unipotent crystal to study the geometric analogues of two symmetries of rectangular tableaux.