There is a class blog, which I'll try to keep up to date with what's happened in class so far, and I'll post exercises there.
I'm keeping the course web page in blog form mainly because it automatically produces an RSS feed. (If that means nothing to you, ask me about it.)

Representation theory of groups can be thought of as the study of homomorphisms into the general linear groups GL_n(C) of invertible matrices (where n varies). One reason to study representation theory of GL_n(C) itself, mapping into GL_m(C) for other m, is that representations of GL_n(C) give representations of any subgroup. Another is that GL_n(C) is the most familiar complex Lie group, and many arguments that work for Lie groups in general (like Jordan canonical form) are more familiar for GL_n(C).

In the first third of the course, we'll construct all the irreducible representations of GL_n, prove that we've found all of them, and study them (e.g. determine their dimensions). One of the basic results here is the characterization of an irrep by its "highest weight". This third of the course is the most algebraic; the basic knowledge required is fluency with tensor products.

The middle third is about the representation ring of GL_n, whose product is defined using the expansion of the tensor product of two irreps again into irreps. This is the most combinatorial, and some of this was the subject of my own research. No background is required.

The last third reprises the first two, but through a geometric lens. The construction of the irreps by hand turns out to be a roundabout approach to the Borel-Weil theorem. The representation ring is connected (in a very subtle way) to intersection theory on Grassmannians. The asymptotics of the tensor product rule are related to questions about the spectrum of the sum of two Hermitian matrices (the first being a quantum mechanical question, and the second its classical mechanical limit). This connection was crucial in my research. In this third, some algebraic geometry would be helpful, but I do plan to introduce all the concepts and definitions I will use. Rather, I would hope that some treat it as an introduction to algebraic geometry motivated by the first two parts of the course.

There are notes on the web page for many of these topics (from a previous course -- they may not mirror exactly how I'll introduce them in this course). I won't be following any other books, though I do recommend Fulton & Harris' Representation Theory.

Graduate and advanced undergraduate students are invited to attend. Knowledge of graduate level abstract algebra will be very helpful.