Lecture 19: Kac-moody Algebra Actions on Categories, I

We have started this class by studying the representation theory of the symmetric group Sn over the complex numbers. We finish by giving a brief introduction to the representation theory of Sn over a field F of positive characteristic p. We will also establish a connection between the representations of ŝlp and those of FSn. This connection was one of motivations to consider Kac-Moody algebra actions on categories. We would like to point out that while the representation theory of Sn in characteristic 0 is a classical and very well understood subject (all representations are completely reducible, the irreducible ones are classified by the Young diagrams, and character formulas are known in some way, at least), the representation theory in characteristic p is very complicated (representations are no longer completely reducible, and, although the classification of the irreducible representations is known, currently, there is not even a conjecture on their characters).