Group Representations and Applications

数学中的群的概念起源于对称性的概念。自然界中的对象或理论构造的对称性可以由组来编码，该组本身携带关于对象结构的重要信息。表示理论允许人们通过群在向量空间上的作用以统一的方式研究群，向量空间模拟了群在应用中的常见行为方式。一个多世纪以来，表象理论一直是数学的中心话题，并在物理和化学中有重要的应用，特别是在量子力学和基本粒子理论中。有限群及其表示在编码理论和密码学中有着重要的应用，在现代计算和数字通信中起着重要的作用。这个项目的目的是增进对有限群表示理论和一些应用的理解。更详细地说，本项目集中于群表示理论及其应用中的几个重要问题。它将数学的不同领域联系在一起，如有限群和代数群、置换群、概率群论、群上同调、组合学、顶点算子代数和代数几何，其中表示论是主要的统一成分。项目中涉及的许多问题都是在研究群表示法时自然而然地提出的，还有一些问题是由应用程序引起的。研究人员将沿着全局-局部原理研究几个问题，包括McKay、Alperin、Brauer的猜想和其他在有限群表示理论中推广和推广经典结果的猜想。研究人员还打算研究低维模表示的分类，并发展特征标级理论，目的是建立有限拟单群的特征标值的强界。这些结果可望用于研究Aschbacher关于子群格的猜想和关于外幂的Kollar-Larsen问题、Miyamoto问题、随机游动和反集中问题、Fuchsian群的表示变种、拟单群的字映射分布和Waring型问题、关于二次上同调群的Holt猜想的改进以及具有特殊性质的拟单群的表示.

The concept of a group in mathematics grew out of the notion of symmetry. The symmetries of an object in nature or of a theoretical construct can be encoded by a group, which itself carries important information about the structure of the object. Representation theory allows one to study groups in a uniform way via their actions on vector spaces, which model common ways groups act in applications. Representation theory has been a central topic in mathematics for more than a century and has important applications in physics and chemistry, particularly in quantum mechanics and the theory of elementary particles. Finite groups and their representations have important applications in coding theory and cryptography, and play an important role in modern computation and digital communications. This project aims to advance understanding in the representation theory of finite groups and in a number of applications. In more detail, this project focuses on several important questions in group representation theory and its applications. It ties together different areas of mathematics, such as finite groups and algebraic groups, permutation groups, probabilistic group theory, group cohomology, combinatorics, vertex operator algebras, and algebraic geometry, with representation theory as the main unifying ingredient. Many of the questions addressed in the project come up naturally in the study of group representations, and others are motivated by applications. The investigator will study several problems along the lines of the global-local principle, including conjectures of McKay, Alperin, Brauer, and others that extend and generalize classical results in the representation theory of finite groups. The investigator also intends to study classification of modular representations of low dimension and to develop a theory of character level, with the goal of establishing strong bounds on character values for finite quasisimple groups. It is anticipated that the results can be applied to study Aschbacher's conjecture on subgroup lattices and the Kollar-Larsen problem on exterior powers, Miyamoto's problem, problems on random walks and anti-concentration, representation varieties of Fuchsian groups, word map distributions and Waring-type problems for quasisimple groups, refinements of Holt's conjecture on second cohomology groups, and representations of quasisimple groups with special properties.