Representations of Finite Groups and Applications

本提案的主要研究领域是有限群的表示理论。数学中群的概念源于对称的概念。自然界或科学中物体的对称性是由一组编码的，其中包含了关于物体本身结构的许多重要信息。群体表示理论允许人们通过他们在向量空间上的行为来研究群体，这些向量空间模拟了他们在现实世界中出现的方式。一个多世纪以来，它一直吸引着数学家，并在量子力学、基本粒子理论、编码理论和密码学中有许多重要的应用，并且预计将继续在现代计算机和数字通信世界中发挥重要作用。本研究将有助于对有限群表示理论的理解和应用。学生的参与将是该项目的科学重要组成部分。本课题主要研究有限群表示理论中的几个问题及其应用。这些问题在群体表征理论中自然出现。其他人则是受到各种应用的激励。PI将研究几个全局-局部问题，包括McKay， Alperin和Brauer的猜想，以及其他一些推广有限群表示理论经典结果的猜想。PI还将继续他的长期项目，发展字符水平理论，建立李型有限群的字符值的强界，并分类低维或具有特殊性质的有限拟单群的模表示。然后，他将应用这些结果在许多应用上取得重大进展，包括局部系统及其单群，Aschbacher关于子群格的猜想，Cayley和McKay图上的随机漫步和Fuchsian群的表示变化，简单群的词映射分布和Thompson猜想，Miyamoto关于顶点算子代数的应用问题，上同调群的界和有限准简单群的表示。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。

The main area of research in this proposal is the representation theory of finite groups. The concept of a group in mathematics grew out of the notion of symmetry. The symmetries of an object in nature or science are encoded by a group, which carries a lot of important information about the structure of the object itself. Group representation theory allows one to study groups via their actions on vector spaces which model the ways they arise in the real world. It has fascinated mathematicians for more than a century, and has had many important applications in quantum mechanics, the theory of elementary particles, coding theory, and cryptography, and is expected to continue to play an important role in the modern world of computers and digital communications. This research will contribute to advances in the understanding and applications of representation theory of finite groups. Student involvement will be a scientifically important component of the project.This project focuses on several problems in the representation theory of finite groups and its applications. Many of these problems come up naturally in the group representation theory. Others are motivated by various applications. The PI will study several global-local problems, including the conjectures of McKay, Alperin, and Brauer, and some other conjectures which generalize classical results in representation theory of finite groups. The PI will also continue his long-term project to develop a theory of character level and to establish strong bounds on character values of finite groups of Lie type, and to classify modular representations of finite quasisimple groups of low dimension or with special properties. He will then apply the results to achieve significant progress on a number of applications, including local systems and their monodromy groups, Aschbacher's conjecture on subgroup lattices, random walks on Cayley and McKay graphs and representation varieties of Fuchsian groups, word map distributions and Thompson's conjecture for simple groups, Miyamoto's problem with applications to vertex operator algebras, and bounds on cohomology groups and presentations of finite quasisimple 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.