# Affine Type $A$ Geometric Crystal on the Grassmannian

## TL;DR

This paper constructs a geometric crystal structure on the Grassmannian that tropicalizes to known combinatorial crystals, revealing connections between geometric symmetries and tableau promotion.

## Contribution

It introduces a new geometric crystal on the Grassmannian linked to affine type A, and relates geometric symmetries to combinatorial promotion and tableau symmetries.

## Key findings

- Constructed a type A_{n-1}^{(1)} geometric crystal on Grassmannian
- Tropicalization yields Kirillov-Reshetikhin crystals for rectangular tableaux
- Established correspondence between geometric symmetries and tableau promotion

## Abstract

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.