Pavel Etingof, MIT – Mathematics Colloquium

Refreshments available in Phillips 330 at 4:00 pm

Title: Representation theory without vector spaces

Abstract: A modern view of representation theory is that it is a study not just of individual representations (say, finite dimensional representations of an affine group scheme G over an algebraically closed field k) but also of the category Rep(G) formed by them. The properties of Rep(G) can be summarized by saying that it is a symmetric tensor category (shortly, STC) which uniquely determines G. It is therefore natural to ask: does the study of STC reduce to group representation theory, or is it more general? In other words, do there exist STC other than Rep(G)? If so, this would be interesting, since one can do algebra in any STC, and in categories other than Rep(G) this would be a new kind of algebra.

The answer turns out to be “yes”, and beautiful examples in characteristic zero were provided by Deligne-Milne in 1981. These very interesting categories are interpolations of representation categories of classical groups GL(n), O(n), Sp(n) to arbitrary values of n in k. Deligne later generalized them to symmetric groups and also to characteristic p, where, somewhat unexpectedly, one needs to interpolate n to p-adic integer values rather than elements of k. All these categories violate an obvious necessary condition for a STC to have any realization by finite dimensional vector spaces (and in particular to be of the form Rep(G)): for each object X the length of the n-th tensor power of X grows at most exponentially with n. We call this property “moderate growth”. So it is natural to ask if there exist STC of moderate growth other than Rep(G).

A remarkable theorem of Deligne (2002) says that in characteristic zero, the answer is “no” if one generalizes groups to supergroups. More precisely, any such category is of the form Rep(G), where G is an affine supergroup scheme. In other words, such a category can be realized in supervector spaces. In particular, algebra in these categories is just the usual one with equivariance under some supergroup G.

However, in characteristic p the situation is much more interesting. Namely, Deligne’s theorem is known to fail in any characteristic p>3 and for p=2 (and also expected to fail for p=3). The simplest exotic symmetric tensor category of moderate growth (i.e., not of the form Rep(G)) for p>3 is the semisimplification of the category of representations of Z/p, called the Verlinde category. For example, for p=5, this category has an object X such that X^2=X+1, so X cannot be realized by a vector space (as its dimension would have to be the golden ratio or its conjugate). I will discuss some aspects of algebra in these categories, in particular failure of PBW theorem for Lie algebras (and how to fix it) and Ostrik’s generalization of Deligne’s theorem in characteristic p. I will also discuss a family of exotic categories in characteristic 2 constructed in my joint work with Dave Benson.