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.

表示论是代数学的一个分支，研究对称性，特别是线性数学结构的对称性，使用可逆矩阵群。有限群的表示理论在其他领域也有许多应用，包括数论和数学物理。在这个项目中的线性结构本身是有限的矩阵群，或更一般的矩阵群，其条目满足整除属性相对于一个固定的素数。几何和组合技术将被用来研究这些群的表示，特别是当表示矩阵本身在有限域中有元素时，更准确地说，本项目的主要目的是：（1）研究Weyl群的表示和有限约化群的幂幺表示的新方法，用Grothendieck群的一个新基;（2）研究了半单群正特征标的特征标公式的一个新形式，（3）在抛物特征标层理论的框架下研究了具有不等参数的Hecke代数，（4）研究了约化群的层，（5）研究了约化群的性质，（6）研究了约化群的性质。最后（5）研究了两个-该奖项反映了NSF的法定使命，并被认为是值得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的支持。