Representations of Finite Groups and Applications

DMS-0600967Pham Huu Tiep 该提案重点关注有限群表示论及其应用中的几个重要问题。其中许多问题在群表示论中自然出现，而其他问题则是由各种应用引起的，特别是在有限本原置换群、李代数、积分格和线性码、组合学、曲线理论和量子信息处理的理论中。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论。研究者打算继续他的长期项目，对低维李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，一方面应提供给定有限群的复数和布劳尔特征的合理性属性，另一方面提供该群的结构之间的联系。随后，他将这两个项目的成果应用到许多应用上，包括有限单群的子群结构、线性表示中群元素的最小多项式、积分格和格拉斯曼设计、本原置换群和曲线有理点的排列、互无偏基和量子信息处理等方面取得了重大进展。本提案的主要研究领域是有限群表示论。数学中的群源于对称的概念。自然界或科学中物体的对称性是由一个群来编码的，这个群携带着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向量空间上的行为来研究群体，该向量空间模拟了它们在现实世界中出现的方式。一个多世纪以来，它一直让数学家着迷，并在物理和化学领域有许多重要的应用，特别是在量子力学和基本粒子理论中。有限群及其表示形式已经在编码理论和密码学中被证明是有价值的，并且有望继续在现代计算机和数字通信世界中发挥重要作用。研究者的研究将在理解有限群表示论方面带来重要进展，并有助于在其许多应用中取得重大进展。

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.