Unipotent elements in algebraic groups and finite groups of Lie type

本课题涉及代数群中单幂元理论中几个密切相关的问题，并应用于代数群和李型有限群的结构理论。本文将基于在Seitz最近的一篇论文中提出的约化群的交换对，提出一种分析单能元中心化器的新方法。将此分析所得的新信息应用于与单幂元典型相关的某一阿贝尔子群的研究，获得关于饱和的新信息，建立单幂元集合的饱和结果，并研究单幂类的新偏序。这些结果将用于研究代数群及其有限类似群的子群结构问题，特别是确定复数上例外群的有限简单子群的共轭类问题的模版本。群的理论有时被描述为对称的数学。它是数学中的一门基础学科，在各种科学领域都有应用。某些类型的群体，那些由李理论产生的群体，一次又一次地出现在广泛不同的背景下，所以建立关于这些群体的一般结果很重要。本课题的目的是获取代数群和相关的李型有限群的此类信息。这些群包含特殊的元素，称为单能元素，它们在一般理论中起着特别重要的作用。尽管这些元素具有根本的重要性，但它们的某些奥秘仍有待了解，一些基本结果尚不令人满意。该项目旨在为这一重要理论发展一种新的方法，澄清现有的结果并建立新的结果。塞茨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。

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.