Intersection cohomology without spaces

Abstract

A beautiful theme in representation theory is that combinatorial and representation-theoretic information about a structure (group, algebra, polytope, etc.) is often captured by the intersection cohomology groups of a related algebraic variety. Intersection cohomology is complicated to compute in general, but for the varieties which arise in this way, there is often a straightforward combinatorial method for computing it. A remarkable feature of these stories is that the graded vector spaces arising in the combinatorial construction are part of a larger family, for which there is no associated variety. In other words, they act as “intersection cohomology without spaces”. 

In this talk, I will explain my favourite example of this phenomenon - the intersection cohomology groups of Schubert varieties and their incarnations as Soergel bimodules. In this example, the special class of groups are Weyl groups, and the larger family are Coxeter groups. 

I will dedicate the first hour to cohomology, with a focus on examples. We will discuss some deep Hodge-theoretic properties of its structure (the hard Lefshetz theorem and the Hodge-Riemann bilinear relations), then I will spend some time motivating intersection cohomology. 

In the second hour, I will describe Soergel’s method for computing the intersection cohomology groups of Schubert varieties in a purely algebraic way. This involves a brief tour of reductive algebraic groups, flag varieties, and Weyl groups. In the final minutes, I’ll explain how this topic touches on my own research by proposing another setting in which “intersection cohomology without spaces” might occur - the settling of real Lie groups.