Geometric and unipotent crystals II: From unipotent bicrystals to crystal bases
Geometric and unipotent crystals II: From unipotent bicrystals to crystal bases
复制标题
几何和单能晶体II：从单能双晶体到晶体底座
DOI：
10.1090/conm/433/08321
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发表时间：
2006
期刊：
影响因子：
--
通讯作者：
D. Kazhdan
中科院分区：
文献类型：
--
作者：
A. Berenstein;D. Kazhdan
对于每个约化代数群G，我们介绍和研究幂幺双晶体作为一个定期版本的双有理几何和幂幺晶体介绍了作者。
幂幺双晶的框架一方面允许系统地研究G中的Bruhat胞及其卷积积等种类，另一方面，给出许多正常柏原晶体的新构造，包括G^\vee-模的那些，其中G^\vee是Langlands对偶群。事实上，我们的晶体碱基类似物（我们称之为与G^\v-模相关的晶体）直接与G^\v-模相关，即，没有量子变形。
本文的主要结果之一是在开Bruhat盒上的正幂幺双晶体的基础上，构造了（Langlands对偶）旗簇坐标环的晶体B_0。
我们的一般热带化程序指定给每个强正幂幺双晶体一个正常的Kashiwara晶体B配备的多重擦除同态B--> B_0和组合中心电荷B--> Z是在所有的晶体运营商不变。
For each reductive algebraic group G we introduce and study unipotent bicrystals which serve as a regular version of birational geometric and unipotent crystals introduced earlier by the authors.
The framework of unipotent bicrystals allows, on the one hand, to study systematically such varieties as Bruhat cells in G and their convolution products and, on the other hand, to give a new construction of many normal Kashiwara crystals including those for G^\vee-modules, where G^\vee is the Langlands dual groups. In fact, our analogues of crystal bases (which we refer to as crystals associated to G^\vee-modules) are associated to G^\vee-modules directly, i.e., without quantum deformations.
One of the main results of the present paper is an explicit construction of the crystal B_0 for the coordinate ring of the (Langlands dual) flag variety based on the positive unipotent bicrystal on the open Bruhat cell.
Our general tropicalization procedure assigns to each strongly positive unipotent bicrystal a normal Kashiwara crystal B equipped with the multiplicity erasing homomorphism B--> B_0 and the combinatorial central charge B--> Z which is invariant under all crystal operators.