Representations of Finite Groups and Applications

DMS-0600967范虎铁讨论了有限群表示论及其应用中的几个重要问题。其中许多问题是在群表示理论中自然而然地出现的，而另一些问题则是由各种应用所驱动的，特别是在有限本原置换群理论、李代数、积分格和线性码、组合学、曲线论和量子信息处理中。它们将不同的数学领域联系在一起，主要的统一成分是表示理论。研究者打算继续他的长期计划，对低维Lie类型有限群的交叉特征表示进行分类。第二个主要项目围绕某些局部-全局问题，一方面应该提供给定有限群的复的理性性质和Brauer特征标与群的结构之间的联系。然后，他将这两个项目的结果应用于一些应用，包括有限单群的子群结构、线性表示中的群元极小多项式、积分格和草曼设计、本原置换群中的错位和曲线的有理点、互无偏基和量子信息处理。数学中的群组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，这个组携带着许多关于物体本身结构的重要信息。呈现理论允许人们通过他们在向量空间上的行为来研究群体，向量空间对他们在现实世界中产生的方式进行建模。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经被证明是有价值的，并有望在现代计算机和数字通信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。

DMS-0600967Pham Huu TiepThis proposal focuses on several important problems in representation theory of finite groups and its applications. Many of these problems come up naturally in the group representation theory, and others are motivated by various applications, particularly in the theory of finite primitive permutation groups, Lie algebras, integral lattices and linear codes, combinatorics, curve theory, and quantum information processing. They tie together different areas of mathematics, with the main unifying ingredient being the representation theory.The investigator intends to continue his long-term project to classify cross characteristic representations of finite groups of Lie type of low dimension. The second main project centers around certain local-global problems, which should provide links between rationality properties of complex and Brauer characters of a given finite groupon the one hand and the structure of the group on the other hand. He then applies the results on these two projects to achieve significant progress in a number of applications, including the subgroup structure of finite simple groups, minimal polynomials of group elements in linear representations, integral lattices and grassmannian designs,derangements in primitive permutation groups and rational points of curves, mutually unbiased bases and quantum information processing.The main area of research in this proposal is therepresentation theory of finite groups. Groups in mathematics grew out of the notion of symmetry. The symmetries of an object in nature orscience are encoded by a group, and this group carries a lot ofimportant information about the structure of the object itself. Therepresentation theory allows one to study groups via their action on vector spaces which models 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 theoryof elementary particles. Finite groups and their representations have already proved valuablein coding theory and cryptography, and are expected to continue to play an important role in themodern world of computers and digital communications. The investigator's research will lead toimportant advances in understanding the representation theory of finite groups and help achieve significant progress in a number of its applications.