Representations of Finite Groups and Applications

本文主要研究有限群表示理论及其应用中的几个重要问题。这些问题中的许多都是在群体表征理论中自然而然地出现的--有些是长期存在的，并发挥着核心作用，另一些则是由各种应用驱动的。该提议将数学的不同领域联系在一起，如有限群和代数群、有限置换群论、群上同调、组合学、算子代数和代数几何，其主要统一成分是表示理论。PI将沿着局部-整体原理研究几个问题，包括Alperin权猜想，Brauer高度零猜想，以及关于有限群的复特征标和Brauer特征标的合理性和可除性的一些进一步的猜想。PI还将继续他的长期项目，对低维有限拟单群的模表示进行分类。然后，他将应用他的结果在一些应用上取得重大进展，包括拟单群的Waring型问题，关于子群格的Aschbacher猜想，关于外幂的Kollar-Larsen问题(及其在代数几何中的应用)，关于有限群的第二上同调群及其表示的Guralnick-Holt猜想，以及具有特殊性质的有限拟单群的表示(通过在有限单群的子群结构中的应用)。数学中的群的概念起源于对称性的概念。自然界或科学中一个物体的对称性是由一个组编码的，而这个组携带着许多关于物体本身结构的重要信息。表象理论允许人们通过他们在向量空间上的行为来研究群体，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，并在物理学和化学中有许多重要的应用，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已被证明是有价值的，并有望在计算机和数字通信的现代世界中继续发挥重要作用。这位研究人员的研究将导致在理解有限群的表示理论方面取得重要进展，并有助于在其一些应用方面取得重大进展。

This proposal focuses on several important problems in representation theory of finite groups and its applications. Many of these problems come up naturally -- some long-standing and playing a central role -- in group representation theory, and others are motivated by various applications. The proposal ties together different areas of mathematics, such as finite groups and algebraic groups, finite permutation group theory, group cohomology, combinatorics, operator algebras, and algebraic geometry, with the main unifying ingredient being the representation theory. The PI will study several problems along the lines of the local-global principle, including the Alperin weight conjecture, Brauer's height zero conjecture, and some further conjectures concerning rationality and divisibility properties of complex and Brauer characters of finite groups. The PI will also continue his long-term project to classify modular representations of finite quasisimple groups of low dimension. He will then apply his results to achieve significant progress on a number of applications, including Waring-type problems for quasisimple groups, Aschbacher's conjecture on subgroup lattices, the Kollar-Larsen problem on exterior powers (with application in algebraic geometry), and the Guralnick-Holt conjecture on second cohomology groups for finite groups and their presentations, and representations of finite quasisimple groups with special properties (with application in the subgroup structure of finite simple groups).The main area of research in this proposal is group representation theory. The concept of a group in mathematics grew out ofthe notion of symmetry. The symmetries of an object in nature or science are encoded by a group, and this group carries a lot of important information about the structure of the object itself. The 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 many important applications in physics and chemistry, particularly in quantum mechanics and in the theory of elementary particles. Finite groups and their representations have already proved valuable in coding theory and cryptography, and are expected to continue to play an important role in the modern world of computers and digital communications. The investigator's research will lead to important advances in understanding the representation theory of finite groups and help achieve significant progress in a number of its applications.