A geometric approach to characters of Hecke algebras
A geometric approach to characters of Hecke algebras
复制标题
赫克代数特征的几何方法
DOI：
--
复制
发表时间：
2022
期刊：
影响因子：
--
通讯作者：
Antonio Nigro
中科院分区：
文献类型：
--
作者：
Alex Abreu;Antonio Nigro
对于一个连通的、单连通的、半单的复代数群G的任何元素和相应Weyl群的一个元素的选择，都有一个相关的Lusztig簇。当G的元素是正则半单时，对应的簇在其(等变)交上同调上同调上带有Weyl群的作用。通过这个作用，我们恢复了相应的Hecke代数的Kazhdan-Lusztig基的一个元素的诱导特征。在A类中，我们证明了一个更精确的命题：这个作用量的Frobenius特征标正是由Kazhdan-Lusztig基元的特征标给出的对称函数。其主要思想是找到这些簇的去单元化的元胞分解，并应用Brosnan-Chow回文判据来确定局部不变循环映射何时是同构的。这恢复了Lusztig关于特征标组的一些结果，并将涉及奇点的Sharesian-Wachs猜想的Brosnan-Chow解推广到非共显性排列。我们还讨论了无差图的内蕴函数、Hecke代数和色拟对称函数之间的关系。
To any element of a connected, simply connected, semisimple complex algebraic group G and a choice of an element of the corresponding Weyl group there is an associated Lusztig variety. When the element of G is regular semisimple, the corresponding variety carries an action of the Weyl group on its (equivariant) intersection cohomology. From this action, we recover the induced characters of an element of the Kazhdan-Lusztig basis of the corresponding Hecke algebra. In type A, we prove a more precise statement: that the Frobenius character of this action is precisely the symmetric function given by the characters of a Kazhdan-Lusztig basis element. The main idea is to find celular decompositions of desingularizations of these varieties and apply the Brosnan-Chow palindromicity criterion for determining when the local invariant cycle map is an isomorphism. This recovers some results of Lusztig about character sheaves and gives a generalization of the Brosnan-Chow solution to the Sharesian-Wachs conjecture to non-codominant permutations, where singularities are involved. We also review the connections between Immanants, Hecke algebras, and Chromatic quasisymmetric functions of indifference graphs.