On the rank varieties and Jordan types of a class of simple modules - Wang, Jialin (Nanyang Technological University, 2024)
Fix a finite group G and an algebraically closed field F of characteristic p. For an FG-module M, the complexity of M is the rate of growth of a minimal projective resolution of M which is a cohomological invariant of M. The rank varieties introduced by Carlson are defined for modules for elementary abelian p-groups and can be extended to modules for G by looking at the restriction to elementary abelian subgroups of G. Moreover, the dimension of the rank variety gives the complexity of the module. In this thesis, I discuss some basic properties of rank varieties and complexities and then review some known results on the complexities of some simple modules for symmetric groups and finite general linear groups.