Support varieties for modules over symmetric groups

Abstract

1.1. In the late 1970s, Alperin [A] defined an invariant called the complexity of a module as a way to relate the modules with the complexes and resolutions that they admit. Several years later, Carlson [Ca1,Ca2] defined affine algebraic varieties corresponding to modules over group algebras. These varieties are subvarieties of the spectrum of the cohomology ring which was earlier described by Quillen [Q]. They are known in present day language as support varieties. It was discovered early on that the complexity of a module is equal to the dimension of the support variety of the module. Geometric methods involving support varieties have played a fundamental role in understanding the interplay between the modular representation theory and cohomology for finite groups. Despite substantial progress in this direction, there have been few explicit computations of support varieties for important classes of modules over certain groups. The goal of this paper is to introduce methods and techniques for computing support varieties for modules over the symmetric group Σd . In the process, we will provide explicit computations of support varieties for certain classes