In a previous paper [J. Combin. Theory Ser. A, 2019], the author 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 ${\mathrm{GL}}_n(\mathbb{C}(\lambda ))$.

在之前的论文中 [ J. Combin。理论系列 A , 2019]，作者在品种上构建了一个几何晶体${\mathbb{X}}_{克}：=\mathrm{格}(克,n)\times {\mathbb{C}}^{\times }$ 它热带化为矩形画面上的仿射晶体结构 $n-克$行。在这个续集中，我们定义并研究了几何 R 矩阵，一个双有理映射$\mathrm{电阻}：{\mathbb{X}}_{{克}_1}\times {\mathbb{X}}_{{克}_2}\to {\mathbb{X}}_{{克}_2}\times {\mathbb{X}}_{{克}_1}$它在成对的矩形画面上热带化为组合R矩阵。我们证明R是几何晶体的同构，并且它满足 Yang-Baxter 关系。在两个表格都有一行的情况下，我们恢复了在许多上下文中出现在文献中的对称群的双有理作用。我们还定义了一个有理函数$乙：{\mathbb{X}}_{{克}_1}\times {\mathbb{X}}_{{克}_2}\to \mathbb{C}$ 由仿射晶体理论热带化为协能函数。

几何R矩阵的大部分性质都来自于它给出了循环群中某个矩阵方程的唯一解的事实${\mathrm{GL}}_n(\mathbb{C}(\lambda ))$.