Unipotent elements in algebraic groups and finite groups of Lie type

李型代数群和有限群中的单能元

• 批准号: 0352749
• 负责人: Gary Seitz
• 金额: $ 30.82万
• 依托单位: University of Oregon Eugene
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-09-01 至 2010-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0352749&HistoricalAwards=false
• 关键词: Unipotent; elements; algebraic; groups; finite

The project concerns several closely related problems in the theory of unipotent elements in algebraic groups, with applications to the structure theory of algebraic groups and finite groups of Lie type. A new approach to the analysis of centralizers of unipotent elements will be carried out, based on commuting pairs of reductive groups as developed in a recent paper of Seitz. New information resulting from this analysis will be applied to the study of a certain abelian subgroup canonically associated with a unipotent element, to obtain new information on saturation, to establish saturation results for certain sets of unipotent elements, and to study a new partial order on unipotent classes. These results will be used to study problems on the subgroup structure of algebraic groups and their finite analogs, in particular the modular version of the problem of determining conjugacy classes of finite simple subgroups of exceptional groups over the complex numbers.The theory of groups is sometimes described as the mathematics of symmetry. It is a fundamental subject within mathematics and has applications in a variety of scientific fields. Certain types of groups, those arising from Lie theory, occur time and again in widely different contexts and so it is important to establish general results about these groups. This project is aimed at obtaining such information for algebraic groups and associated finite groups of Lie type. These groups contain special elements, called unipotent elements, which play a particularly important role in the general theory. In spite of their fundamental importance, certain mysteries regarding these elements remain to be understood and some basic results are in an unsatisfactory state. The project is aimed at developing a new approach to this important theory, clarifying existing results and establishing new results. Seitz is in excellent position to address these problems and has recently obtained new results in the area.

该项目涉及代数群中单能元理论中的几个密切相关的问题，以及在代数群和李型有限群的结构理论中的应用。 将基于 Seitz 最近的一篇论文中开发的还原群交换对，采用一种新的方法来分析单能元素的中心化子。 该分析产生的新信息将应用于研究与单能元素规范相关的某个阿贝尔子群，以获得有关饱和的新信息，建立某些单能元素集的饱和结果，并研究单能类的新偏序。这些结果将用于研究代数群及其有限类似物的子群结构问题，特别是确定复数上异常群的有限简单子群的共轭类问题的模版本。群论有时被描述为对称性数学。 它是数学中的一门基础学科，在各种科学领域都有应用。某些类型的群，即由李理论产生的群，在截然不同的环境中一次又一次地出现，因此建立关于这些群的一般结果非常重要。 该项目旨在获取代数群和相关李型有限群的信息。这些基团包含特殊元素，称为单能元素，它们在一般理论中发挥着特别重要的作用。 尽管它们具有根本重要性，但有关这些元素的某些奥秘仍有待理解，并且一些基本结果仍不令人满意。 该项目旨在为这一重要理论开发一种新方法，澄清现有结果并建立新结果。 Seitz在解决这些问题方面处于有利地位，并且最近在该领域取得了新的成果。