University of Sydney Algebra Seminar

Ben Webster (University of Virginia)

Full Schedule of Lectures:

1) Wednesday June 3, 4-5:30pm, Carslaw 829 (Access Grid Room)
2) Wednesday June 10, 4-5:30pm, Carslaw 829 (Access Grid Room)
3) Wednesday June 17, 4-5:30pm, Carslaw 829 (Access Grid Room)
4) Wednesday June 24, 4-5:30pm, Carslaw 829 (Access Grid Room)

Representation theory through the lens of categorical actions

Representation theorists (and other mathematicians) have come to appreciate that a vector space or set having self-maps that enjoy certain relations can tell one a great deal about the object being studied. These talks will be about the categorified version of this principle: that a category is much easier to study when it carries an action of functors which satisfy certain relations (where, of course, one must be more careful about what relations means).

In a remarkable self-similarity, many categories that are of great interest to representation theorists are themselves representations of a more complicated type. Specifically, they are special cases of a more general notion of an "action of a Lie algebra on a category." Thus, the entire representation category of all symmetric groups over a finite field is actually a representation of an affine Lie algebra. This is a very powerful notion, which both leads us to unexpected structures (such as gradings on algebras without an obvious homogeneous presentation) and to a new perspective on older results (for example, the identification of decomposition numbers with Kazhdan-Lusztig polynomials).

I'll start by discussing the case of the symmetric group and general linear groups over a finite field, and expand outward to less familiar objects like categories O for Lie algebras and Cherednik algebras. In further lectures, I'll discuss other applications of these ideas to areas like canonical bases, algebraic geometry and knot homology.