再论典范基的组合学：A型
Combinatorics of canonical bases revisited: Type A
摘要：
我们提出了一种新的方法来研究组合的几个参数化的典型基地。本文研究了$A$型李代数。使用几何对象称为菱形平铺，我们推导出一个“交叉公式”来计算Lusztig数据的晶体运营商的行动，为任意减少字的最长外尔群元素。我们提供了这个结果的以下三个应用。利用Berenstein-Fomin-Zelevinsky的热带腔室模型，我们证明了MV多面体上晶体结构的Anderson-Mirković猜想的一个增强版本。我们建立柏原的字符串和Lusztig的参数化之间的对偶性，揭示了它们中的每一个是由晶体结构的其他控制。我们确定了由Berenstein-Kazhdan和Gross-Hacking-Keel-Kontsevich分别定义的$SL_n$的幂幺根的势函数，其中一个函数来自Lusztig数据的晶体结构。
英文摘要：
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" to compute the actions 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 $SL_n$ defined by Berenstein-Kazhdan and Gross-Hacking-Keel-Kontsevich, respectively, with a function arising from the crystal structure on Lusztig data.
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