Representations of Finite Groups and Applications

DMS-0600967 Pham Huu Tiep这一建议集中于有限群表示论及其应用中的几个重要问题。这些问题中的许多自然出现在群表示论中，而其他问题则受到各种应用的启发，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等理论中。他们联系在一起的不同领域的数学，与主要的统一成分是表示theory.The调查打算继续他的长期项目，以分类交叉特征表示有限群的李型低维。第二个主要项目围绕某些局部-全局问题，这应该提供一个给定的有限群的复和Brauer特征的合理性属性之间的联系，一方面和另一方面的组的结构。然后，他应用这两个项目的结果，以取得重大进展的一些应用，包括子群结构的有限简单的群体，最小多项式的群元素在线性表示，积分格和格拉斯曼设计，乱序在原始置换群和合理的点的曲线，相互无偏基和量子信息处理。该计划的主要研究领域是有限群的表示理论。数学中的群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，向量空间模拟了它们在真实的世界中出现的方式。它已经吸引了数学家们超过一个世纪，并在物理和化学，特别是在量子力学和基本粒子理论中有许多重要的应用。有限群及其表示已经在编码理论和密码学中被证明是有价值的，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.