Time

May 27 2008 - 4:30pm - 5:30 pm

Location

Scott Lab 241

Speaker

Julianna Tymoczko (Iowa)

Seminar Website

http://www.math.ohio-state.edu/~lchen/seminar.html

Abstract

While exploring his celebrated homology theory for tangles, Khovanov developed a family of rings that are invariants of tangle cobordisms. He made a remarkable discovery: the center of each ring is isomorphic to the cohomology of a Springer fiber. Springer fibers are subvarieties of the flag variety; amazingly, their cohomology carries a natural action of the symmetric group, and the top dimensional cohomology is an irreducible representation. Unfortunately, traditional geometric and topological constructions of the Springer representations use high-powered technical tools that make explicit calculations near impossible. We use Khovanov's results to give an explicit geometric presentation of some Springer representations as well as an explicit combinatorial presentation in terms of certain braid actions on noncrossing matchings. As an application, we explicitly compute Springer representations outside of the top-degree case. We obtain the new result that for the so-called two-row Springer fibers, each graded part of the cohomology is an irreducible representation. This is joint work with Heather Russell.