Unipotent elements in algebraic groups and finite groups of Lie type

该项目涉及代数群中的幂等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。在Seitz最近的一篇论文中提出了一种分析幂等元素中心化子的新方法，该方法基于还原群的交换对。由这一分析得到的新信息将被应用于研究与一个幂等元规范相关的某个交换子群，得到关于饱和的新信息，建立某些幂等元集合的饱和结果，并研究关于酉元类的一个新的偏序。这些结果将被用来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单子群的共轭类的模版问题。群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，那些产生于李氏理论的群，在截然不同的背景下一次又一次地出现，因此建立关于这些群的一般结果是重要的。这个项目的目的是获得关于李型代数群和相关有限群的此类信息。这些群包含特殊的元素，称为单幂等元素，它们在一般理论中扮演着特别重要的角色。尽管它们具有根本性的重要性，但关于这些因素的某些谜团仍有待理解，一些基本结果也处于不令人满意的状态。该项目的目的是为这一重要理论开发一种新的方法，澄清现有的结果并确定新的结果。塞茨处于解决这些问题的绝佳地位，最近在这一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.