The geometric $R$-matrix for affine crystals of type $A$

In [Frieden, arXiv:1706.02844], we constructed a geometric crystal on the variety $\mathbb{X}_{k} := {\rm Gr}(k,n) \times \mathbb{C}^\times$ which tropicalizes to the affine crystal structure on rectangular tableaux with $n-k$ rows. In this sequel, we define and study the geometric $R$-matrix, a birational map $R : \mathbb{X}_{k_1} \times \mathbb{X}_{k_2} \rightarrow \mathbb{X}_{k_2} \times \mathbb{X}_{k_1}$ which tropicalizes to the combinatorial $R$-matrix on pairs of rectangular tableaux. We show that $R$ is an isomorphism of geometric crystals, and that it satisfies the Yang--Baxter relation. In the case where both tableaux have one row, we recover a birational action of the symmetric group that has appeared in the literature in a number of contexts. We also define a rational function $E : \mathbb{X}_{k_1} \times \mathbb{X}_{k_2} \rightarrow \mathbb{C}$ which tropicalizes to the coenergy function from affine crystal theory. Most of the properties of the geometric $R$-matrix follow from the fact that it gives the unique solution to a certain equation of matrices in the loop group ${\rm GL}_n(\mathbb{C}(\lambda))$.