Math 577: Introduction to representation theory

Ron Irving

Autumn 1999, MWF 9:30-10:20

The course will begin with an introduction to some standard topics in the representation theory of groups, focusing in particular on finite groups. We'll look at representations, irreducibility, complete reducibility, examples, characters, character formulas, and sample calculations. Tensor products and induction will then be introduced, both as theoretical tools and as a means to simplify calculations. Special attention will be given to the two smallest non-abelian simple groups, the alternating group of order 60 and the projective special linear group of order 168.

There are many approaches to the material above. A group-theoretic approach allows one to extend these ideas from finite groups to the broader class of compact groups, including orthogonal and unitary groups. I will take this approach, allowing us to take a short look at how the results extend to compact groups. Alternatively, a ring-theoretic approach is the most efficient means to obtain additional information about finite group representations. I will return to finite groups, introduce this approach, and obtain some of these additional consequences. If time remains, I will conclude the course with an introduction to the invariant theory of finite groups.

 The first-year algebra course, Math 504-5-6, is the natural prerequisite for this course. In Winter, 2000, Professor McGovern is teaching Math 578 at the same time. Math 577 and Math 578 are independent of each other, but they are thematically related and form a natural two-quarter sequence.