Representations of finite reductive groups, character sheaves and theory of total positivity

有限约简群的表示、特征轮和总正性理论

• 批准号: 2153741
• 负责人: George Lusztig
• 金额: $ 20.72万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-07-01 至 2025-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2153741&HistoricalAwards=false
• 关键词: Representations; finite; reductive; groups; character

Representation theory is a branch of algebra studying symmetries, especially symmetries of linear mathematical structures, using groups of invertible matrices. Representation theory of finite groups has numerous applications to other areas, including number theory and mathematical physics. In this project the linear structures are themselves finite matrix groups, or more generally matrix groups whose entries satisfy divisibility properties with respect to a fixed prime number. Geometric and combinatorial techniques will be brought to bear to study representations of these groups, especially in the important case when the representing matrices themselves have entries in a finite field.More precisely, the central aims of this project are to (1) investigate a new approach to representations of Weyl groups and unipotent representations of finite reductive groupsin terms of a new basis of the Grothendieck group; (2) investigate a new formulation of the character formula for semisimple groups in positive characteristic; (3) study Hecke algebras with unequal parameters in the framework of the theory of parabolic character sheaves; (4) study strata of a reductive group; and finally (5) investigate new W-graphs associated to involutions in two-sided cells of a Coxeter groups.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

表示理论是使用可逆矩阵组的代数研究对称性的代数分支，尤其是线性数学结构的对称性。有限群体的表示理论在其他领域有许多应用，包括数字理论和数学物理学。在此项目中，线性结构本身是有限的矩阵组，或更一般的矩阵组，其条目满足固定质量数量的分裂性能。几何技术和组合技术将被带到研究这些群体的表示形式中，尤其是在代表矩阵本身具有有限领域的条目时，尤其是，更确切地说，该项目的核心目的是（1）调查一种新的weyl群体代表性的方法，并针对Weyl群体的代表和无能为力的小组，对绿地的有限型组成式的有限型组成部分，这是一个新的款项。 （2）调查在积极特征中为半圣经群体的字符公式的新表述； （3）在抛物线特征束理论框架内研究Hecke代数的，具有不平等的参数； （4）还原群的研究阶层；最后（5）调查了与Coxeter组的双面细胞相关的新的W指法。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛的影响评估标准通过评估来支持的。