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

In [Frieden, arXiv:1706.02844], we constructed a geometric crystal on the variety ${\mathbb{X}}_k:=\mathrm{G}\mathrm{r}(k,n)\times {\mathbb{C}}^{\times }$ 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 ${\mathrm{G}\mathrm{L}}_n(\mathbb{C}(\lambda ))$.