Geometric Lifting of Affine Type A Crystal Combinatorics

[摘要] In the first part of this thesis, we construct a type $A_{n-1}^{(1)}$ geometric crystal on the variety $mathbb{X}_k := {rm Gr}(k,n) times mathbb{C}^times$, and show that it tropicalizes to the disjoint union of the Kirillov--Reshetikhin crystals corresponding to rectangular semistandard Young tableaux with $n-k$ rows. A key ingredient in our construction is the $mathbb{Z}/nmathbb{Z}$ symmetry of the Grassmannian which comes from cyclically shifting a basis of the underlying vector space. We show that a twisted version of this symmetry tropicalizes to combinatorial promotion.In the second part, 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 the birational $R$-matrix of Yamada and Lam--Pylyavskyy. 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 ${rm GL}_n(mathbb{C}(lambda))$.

[发布机构] University of Michigan

[效力级别] geometric crystals

[关键词] crystals;geometric crystals;combinatorial $R$-matrix;Mathematics;Science;Mathematics