We initiate a new approach to the study of the combinatorics of several parametrizations of canonical bases. In this work we deal with Lie algebras of type A. Using geometric objects called rhombic tilings we derive a “Crossing Formula” for the action of the crystal operators on Lusztig data for an arbitrary reduced word of the longest Weyl group element. We provide the following three applications of this result. Using the tropical Chamber Ansatz of Berenstein–Fomin–Zelevinsky we prove an enhanced version of the Anderson–Mirković conjecture for the crystal structure on MV polytopes. We establish a duality between Kashiwara’s string and Lusztig’s parametrization, revealing that each of them is controlled by the crystal structure of the other. We identify the potential functions of the unipotent radical of a Borel subgroup of \(SL_n\) defined by Berenstein–Kazhdan and Gross–Hacking–Keel–Kontsevich, respectively, with a function arising from the crystal structure on Lusztig data.

我们启动了一种新方法来研究规范基础的几种参数化的组合。在这项工作中，我们处理类型A 的李代数. 使用称为菱形拼贴的几何对象，我们推导出一个“交叉公式”，用于晶体算子对 Lusztig 数据的作用，用于最长 Weyl 组元素的任意简化字。我们提供了这个结果的以下三个应用。使用 Berenstein-Fomin-Zelevinsky 的热带室 Ansatz，我们证明了 MV 多胞体晶体结构的 Anderson-Mirković 猜想的增强版本。我们在 Kashiwara 的弦和 Lusztig 的参数化之间建立了对偶性，揭示了它们中的每一个都受另一个的晶体结构控制。我们确定了\(SL_n\)的 Borel 子群的单能根的势函数 分别由 Berenstein-Kazhdan 和 Gross-Hacking-Keel-Kontsevich 定义，函数来自 Lusztig 数据的晶体结构。