Representations of Finite Groups and Applications

有限群的表示及其应用

• 批准号: 0964957
• 负责人: Pham Tiep
• 金额: $ 2.99万
• 依托单位: University of Arizona
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-01 至 2011-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0964957&HistoricalAwards=false
• 关键词: Representations; Finite; Groups; Applications

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.

DMS-0600967PHAM HUU TIEP TIEPTHIS提案重点介绍了有限群体及其应用的代表理论中的几个重要问题。这些问题中的许多人自然出现在小组代表理论中，而其他问题则是由各种应用的动机，尤其是在有限原始置换群体的理论中，谎言代数，整体晶格和线性代码，组合仪，曲线理论和量子信息处理。它们将不同的数学领域联系在一起，主要的统一要素是代表理论。研究人员打算继续他的长期项目，以对低维度的有限层类型类型的有限群体进行分类。第二个主要项目围绕某些本地全球问题，该问题应提供给定有限groupon的复杂性和brauer字符的合理性属性之间的联系，另一方面是该组的结构。然后，他将结果应用于这两个项目，以在许多应用中取得重大进展，包括有限简单组的亚组结构，线性表示中的组要素的最小多项式，整体晶格和格拉曼尼亚的设计，曲线的原始置换群体和曲线的扰民的扰动，曲线的cortrational compriation compriase base base base base和量级的信息。数学的群体是从对称的概念中得出的。对象在自然界中的对称性是由一个组编码的，并且该组具有有关对象本身结构的许多重要信息。其说明理论允许一个人通过对向量空间的行动来研究小组，这些矢量空间模拟了它们在现实世界中的出现方式。它使数学家着迷了一个多世纪，并且在物理和化学方面有许多重要的应用，尤其是在量子力学和基本颗粒理论中。有限的群体及其表示形式已经证明是有价值的编码理论和密码学，并有望继续在计算机和数字通信世界中发挥重要作用。研究者的研究将导致在理解有限群体的表示理论方面的重要进展，并有助于在其许多应用中取得重大进展。