Constructing basic algebras for the principal block of sporadic simple groups

Abstract

This dissertation describes an algorithm for constructing the basic algebra Morita equivalent to the principal block of certain group algebras. This algorithm uses the method of condensation as it is described in [Lux97]. Using an intermediate condensation subalgebra allows for the construction of the projective indecomposable modules required to realize the basic algebra. The group algebras we are concerned with here are for sporadic groups in characteristic dividing the order of the group. In particular, the basic algebra for the principal block of the Higman-Sims group in characteristic 2 is completed and seven of the thirteen projective indecomposable modules for the Mathieu group M24 are constructed. In addition to these algebras, we have also computed the basic algebras for many alternating and symmetric groups.