Group Representations and Applications

本文主要讨论有限群表示论及其应用中的几个重要问题。它将不同的数学领域联系在一起，例如有限群和代数群，有限置换群理论，群上同调，组合学和有限几何，代数几何和弦理论，主要的统一成分是表示论。在提案中解决的许多问题自然出现-一些长期存在，并发挥核心作用-在群表示理论，和其他人的动机是各种重要的应用。PI将沿着局部-全局原理研究几个问题，包括Brauer的零高度猜想和一些关于复形的合理性和整除性以及有限群的Brauer特征的问题。 PI还将继续他的长期项目，对低维Lie型有限群的交叉特征表示进行分类。然后，他将应用他的结果，以寻求重大进展的一些应用，包括Kollar-Larsen问题的线性群与元素的有界年龄（在代数几何和弦理论中的应用），Ore猜想关于单群中的代数子，加强Holt猜想关于有限群的第二上同调群的维数及其表示，以及具有特殊性质的有限拟单群的表示（以及在有限单群的子群结构中的应用）。本建议的主要研究领域是群表示理论。数学中群的概念是从对称性的概念发展而来的。在自然界或科学中，物体的对称性是由一个群编码的，这个群携带了许多关于物体本身结构的重要信息。表示论允许人们通过向量空间上的行为来研究群体，向量空间模拟了群体在真实的世界中出现的方式。它吸引了数学家们超过世纪，并在物理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经被证明是有价值的，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。研究人员的研究将导致理解有限群的表示理论的重要进展，并有助于在其应用中取得重大进展。

This proposal focuses on several important problems in representation theory of finite groups and its applications. It ties together different areas of mathematics, such as finite groups and algebraic groups, finite permutation group theory, group cohomology, combinatorics and finite geometry, algebraic geometry, and string theory, with the main unifying ingredient being the representation theory. Many of the problems addressed in the proposal come up naturally -- some long-standing and play a central role -- in the group representation theory, and others are motivated by various important applications. The PI will study several problems along the lines of the local-global principle, including Brauer's height zero conjecture and some 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 cross characteristic representations of finite groups of Lie type of low dimension. He will then apply his results to seek significant progress on a number of applications, including the Kollar-Larsen Problem on linear groups with elements of bounded age (with applications in algebraic geometry and string theory), the Ore Conjecture on commutators in simple groups, a strengthening of Holt's Conjecture on the dimension of the second cohomology group 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 the group representation theory. 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, 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 had 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.