Subgroups and Representations of Algebraic Groups and Associated Finite Groups

Seitz，9610334所提出的工作是关于李型单代数群和有限群的子群结构和表示的一些问题。主要问题是完成例外代数群的极大子群的分类，确定有有限多个双陪集的单代数群的极大子群对，以及建立有限群的提升表示为相应代数群表示的结果。其他问题涉及不可约模中的权空间的维度的界，以及有限群在伴随模上的不可约作用问题。群是代数对象，只要存在对称性就会出现，研究群的一般性质及其表示是很自然的。许多问题可以归结为对单群的子群的分析。然而，这样的分析可能非常困难，并最终与表征理论中的基本问题联系在一起。Seitz有一个长期的研究计划，旨在系统地研究一些最有趣的单群族、单代数群及其有限类似群的子群和表示。

SEITZ, 9610334 The proposed work concerns a number of problems on the the subgroup structure and representations of simple algebraic groups and finite groups of Lie type. The main problems are to complete the classification of the maximal subgroups of exceptional algebraic groups, to determine pairs of maximal subgroups of simple algebraic groups for which there are finitely many double cosets, and to establish results on lifting representations of finite groups to representations of corresponding algebraic groups. Additional problems concern bounds for dimensions of weight spaces in irreducible modules and a problem concerning finite groups acting irreducibly on adjoint modules. Groups are algebraic objects that appear whenever there is symmetry and it is natural to study general properties of groups as well as their representations. Many problems can be reduced to the analysis of subgroups of simple groups. However, such an analysis can be extremely difficult and is ultimately linked to fundamental problems in representation theory. Seitz has a long standing research program aimed at systematically studying the subgroups and representations of some of the most interesting families of simple groups, the simple algebraic groups and their finite analogs.