Modular Representation Theory and Phantom Maps

In this talk, I will introduce and motivate the main objects of study in modular representation theory, which leads one to consider the stable module category. Perhaps surprisingly, one can do homotopy theory in this setting, as the stable module category has the structure of a stable model category. Therefore, there is a very rich interplay between modular representation theory and stable homotopy theory, which I will try to illustrate. In the spirit of Halloween, one of the ideas I will talk about is the notion of phantom maps. 1. (Modular) Representation Theory For the entirety of this talk, let k be a field of positive characteristic p, and let G be a finite group such that p |G|. We would like to look at the representations ρ̂ : G → Endk(V ). We can then extend this to a group algebra representation ρ : kG→ Endk(V ), which is the same as looking at modules over the group algebra kG. This is why we consider the category of modules over kG. Why is this different from ordinary/characteristic 0 representation theory? For one thing, Maschke’s theorem does not apply: Theorem 1.1 (Maschke). The group algebra kG is semisimple (Cartesian product of simple subalgebras) iff the characteristic of k does not divide the order of G. Remark 1.2. The proof involves dividing by |G|. This in particular tells us things about projective modules: Proposition 1.3. The following are equivalent definitions for a module P being projective: (i) For every surjective map M → N and every map P → N , there exists a map from P →M . M