Title: A combinatorial and geometric construction of the Springer representation
Abstract: Springer fibers are algebraic varieties whose cohomology carries a natural action of the symmetric group; 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 calculations near impossible.
We use recent results of Khovanov about a family of knot invariants 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 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.