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.]

The tensor product construction of the induced representation? I did not really understand induction (that is, I didn’t understand why a thing such as induction ought to exist) until I saw it from the module point of view.

You can also use the group algebra to cast projection operators in the more general framework of central idempotents of a ring.

The things that most convinced me of the importance of group algebras (or, for Lie groups, the universal enveloping algebra of the Lie algebra) are two constructions crucial in classifying representations:

1. The quadratic Casimir, or more generally, center of the universal enveloping algebra. These operators on a representation are critical tools for understanding it, hard to understand them just in terms of representations as homomorphisms.

2. The construction of Hecke algebras as biinvariant elements in the group algebra. Again, if you want to classify representations, a Hecke algebra is often the way to do it, and these are best understood in terms of the group algebra.

#1: sure, the tensor product helps understand induction — that’s what I am going to explain — but it is not needed, and I would say that writing down the function-on-the-group model is more concrete (I wrote the whole shebang using functions: Frobenius reciprocity, transitivity, dimension formula, projection formula…)

Moreover, when dealing, say, with infinite-dimensional unitary or admissible representations of reductive groups, as occur in the theory of automorphic forms, how to use the tensor product becomes much less obvious.

#2: Those are two excellent examples, of course, which I hope to mention at least briefly during the semester, but they are already quite advanced from the point of view of this course.

(For context, I do not intend to assume that the students have had any extensive prior exposure to the tensor product.)

Just an idle thought: how about working over other fields?

e.g. it seems like a good question

to ask starting students to classify representations of a cyclic group of order k over a finite field. They should be able to see “by hand” that this has similar structure to factorizing x^k-1. The latter is clearly a problem about rings, and is a shadow of working in the group algebra.

I think that’s a good example indeed. (Especially since I realized that one can prove the other property I mentioned in homomorphism-language without much trouble).

The whole rationality question for characters/representations does seem to be really better in terms of algebras; I am shamefully unfamiliar with the fine details of that area…

Looking forward to knowing something about “category”. I studied(=! understand) a little about categorical products of topological spaces.