Maximal subgroups, 1-cohomology, and reduced standard modules

Prof. Leonard Scott (University of Virginia)

Abstract: I will review some components of the theory of maximal subgroups of finite groups. Maximal subgroups correspond bijectively to primitive permutation representations,the simple objects in the category of permutation actions. Though these actions are nonlinear, it has been known for some years that, with the classification of finite simple groups in hand, irreducible linear representations of algebraic and finite groups of Lie type play a controlling role in the theory, both in themselves and in their 1-cohomology. I will review this theory, some 1-cohomology conjectures of Bob Guralnick, and present some new positive evidence for them, or at least for the viewpoint they represent. The approach is through further study by CPS of a class of modules studied by Zongzhu Lin (a former ECNU student), which are reductions mod p of lattices in irreducible modules over the Lusztig quantum group. Lusztig had also noticed these modules, but Lin's work gives them a second interpretation which does not depend on quantum groups. We are able to obtain very general results on all degree Ext groups between these modules, especially strong in the degree 1 case, yielding new progress on the main Guralnick conjecture.

Report time: 2007-07-09 16:10-17:10
Report place: Science Building A508