The Spring semester at ETH is starting next week, and I will be teaching an introductory course on representation theory (for third-year and later students). I am looking forward to it, since this is the first time I teach a class identified as “algebra” (except for linear algebra, of course).
My lecture notes will be available as I prepare them (the first chapter will be ready soon, at least in draft form) and it will be seen that (partly because of my own bias) I think of a representation of a group as a homomorphism , and not as modules over the group algebra. I certainly intend to mention the latter approach at some point (indeed, I have in preparation a long “Variants” chapter, where I mention also, e.g., topological representations, unitary representations, permutation representations, projective representations, Lie algebra representations, etc, with basic examples), but I find it useful to go through the elementary theory of representations completely “by hand”, for a course at this level. In some sense, this is because this is what I can do best, with my own knowledge; but also, going through these basic facts purely with the tools and language of representations-as-homomorphisms does provide excellent opportunities to start seeing various ways of using representation theory (I will for instance prove Burnside’s irreducibility criterion and the linear independence of matrix coefficients using my understanding of Burnside’s original arguments). And I do intend to use this course to introduce some functorial language to the students, and I feel that abstract nonsense will be quite appealing after trying to make sense of the confusion that may arise from proving the transitivity formula for induction strictly using the “subset of functions on the group” model for induced representations…
Here’s a question about the module-over-the-group-algebra approach: what is the first (or simplest, or most fundamental) argument where it is better? The one I can think of for the moment — after going through the first chapter of my notes — is that it seems to give the simplest way to argue that a semisimple representation always contains an irreducible subrepresentation. (And of course this is only relevant for infinite-dimensional representations, since otherwise one can argue using dimension.)
[Update (23.2.2011): the first chapters are now online.]