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

Abstract

In [Frieden, arXiv:1706.02844], we constructed a geometric crystal on the variety ${\mathbb{X}}_k:=\mathrm{Gr}(k,n)\times {\mathbb{C}}^{\times }$ 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 $R:{\mathbb{X}}_{k_1}\times {\mathbb{X}}_{k_2}\to {\mathbb{X}}_{k_2}\times {\mathbb{X}}_{k_1}$ 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 $E:{\mathbb{X}}_{k_1}\times {\mathbb{X}}_{k_2}\to \mathbb{C}$ 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 ${\mathrm{GL}}_n(\mathbb{C}(\lambda ))$.

arXiv comment54 pages. (v2) Added more background in Introduction; fixed typos