Breadcrumb
Modular representation theory of finite groups
Supervisor: Jeremy Rickard
Theme: Representation Theory
A representation of a finite group G is just an action of G by linear maps on a vector space V, or in other words a group homomorphism from G to the group GL(V) of invertible linear endomorphisms of V. You may have come across representations over the complex numbers in an undergraduate course. In this case, the problem of classifying representations reduces to finding the (finitely many) irreducible representations, since a general representation is a sum of these. The word 'modular' refers to the study of representations over fields of non-zero characteristic p, where p divides the order of the group G. Here matters are much more complicated, as it is no longer true that every representation is a sum of irreducibles, and it is only in the simplest cases that a complete classification has been found. However, this added complexity has the compensating advantage that more powerful techniques can be brought to bear. Indeed, much of the most important current work on modular representation theory involves ideas related to homological algebra and algebraic topology.