The study of elementary abelian groups is a central topic in the representation theory of finite groups. For example, if G is a finite group and k is a field of characteristic p, then a theorem of Chouinard tells us that a finitely-generated kG-module is projective if and only if it is projective when restricted to any elementary abelian subgroup of G. Hence detecting the projectivity of a module in this case reduces to understanding the behavior of modules for elementary abelian groups. A major problem in studying the modules for an elementary abelian group is that its group algebra almost always has wild representation type, that is, its modules are impossible to classify. Hence we are forced to restrict our attention to certain classes of these modules. In this talk I will introduce an interesting class of modules called modules of constant Jordan type, giving some motivation for the subject and a brief introduction to the algebraic geometry used in the theory. I will then discuss some conjectures about which Jordan types are realizable as the Jordan type of a finitely-generated module for an elementary abelian group.
Modules of Constant Jordan Type I