Subgroups and Representations of Algebraic Groups and Associated Finite Groups

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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.