Representations of Finite Groups and Applications

本文主要讨论有限群表示理论中的几个重要问题及其应用。这些问题中有许多是自然出现的——有些是长期存在的，在群体表征理论中起着核心作用，而另一些则是由各种应用驱动的。该建议将数学的不同领域联系在一起，如有限群和代数群、有限置换群理论、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-全局原理研究几个问题，包括Alperin权猜想，Brauer高度零猜想，以及关于有限群复和Brauer特征的合理性和可整除性的进一步猜想。PI还将继续他的长期项目，分类低维有限拟单群的模表示。然后，他将把他的结果应用到许多应用中，包括拟单群的waring型问题，子群格上的Aschbacher猜想，外幂的kolar - larsen问题（及其在代数几何中的应用），有限群的第二上同群的guralnicki - holt猜想及其表示。以及具有特殊性质的有限拟单群的表示（及其在有限单群子群结构中的应用）。本提案的主要研究领域是群体表征理论。数学中群的概念源于对称的概念。自然界或科学中一个物体的对称性是由一个群体编码的，这个群体携带了很多关于物体本身结构的重要信息。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，在物理和化学，特别是在量子力学和基本粒子理论中有许多重要的应用。有限群及其表示在编码理论和密码学中已经被证明是有价值的，并且有望在现代计算机和数字通信世界中继续发挥重要作用。研究者的研究将在理解有限群的表示理论方面取得重要进展，并有助于在许多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.