Affine Type $A$ Geometric Crystal on the Grassmannian

We construct a type $A_{n-1}^{(1)}$ geometric crystal on the variety ${\rm Gr}(k,n) \times \mathbb{C}^\times$, 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 $\mathbb{Z}/n\mathbb{Z}$ 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 ${\rm GL}_n(\mathbb{C}(\lambda))$ 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.