The geometric R-matrix for affine crystals of type A

In a previous paper [J. Combin. Theory Ser. A, 2019], the author constructed geometric crystal on variety Xk:=Gr(k,n)×C× which tropicalizes to affine structure rectangular tableaux with n−k rows. this sequel, we define and study R-matrix, birational map R:Xk1×Xk2→Xk2×Xk1 combinatorial R-matrix pairs of tableaux. We show that R is an isomorphism crystals, it satisfies Yang–Baxter relation. case where both have one row, recover action symmetric group has appeared in literature number contexts. also rational function E:Xk1×Xk2→C coenergy from theory. Most properties follow fact gives unique solution certain equation matrices loop GLn(C(λ)).