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.

表示论是代数的一个分支，研究对称，特别是线性数学结构的对称，使用可逆矩阵群。有限群的表示理论在数论和数学物理等领域有着广泛的应用。在这个项目中，线性结构本身是有限矩阵群，或者更一般地说，矩阵群的项满足对固定素数的可整除性。几何和组合技术将用于研究这些群的表示，特别是在表示矩阵本身在有限域中有条目的重要情况下。更准确地说，这个项目的中心目标是：(1)根据Grothendieck群的新基础，研究Weyl群的表示和有限约化群的单幂表示的新方法；(2)研究了具有正特征的半单群的特征公式的一个新的表述；(3)在抛物特征束理论的框架下研究不等参数Hecke代数；(4)研究还原群地层；最后(5)研究了与Coxeter组双侧细胞内翻相关的新w图。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。