Low Dimensional Representations of Finite Groups and Conjectures of Serre and Larsen

Robert Guralnick, University of Southern California

By Maschke's theorem, every finite dimensional representation of a finite group in characteristic zero is completely reducible. This fails in positive characteristic. We will show that for small enough representations, one can still deduce complete reducibility. A special case of this is a conjecture of Serre. We will also discuss a conjecture of Michael Larsen about the decomposition of small tensor powers of irreducible modules. The key to both results is to study "small" representations of finite simple groups.