Richard Stanley introduced the chromatic symmetric function of a graph, which contains at least as much information about the graph as the well-studied chromatic polynomial. In joint work with Michelle Wachs, we introduced a refinement of Stanley's chromatic symmetric function, called the chromatic quasisymmetric function. When G is the incomparability graph of a unit interval order, the chromatic quasisymmetric function of G turns out to be a symmetric function, and is Schur positive. This means that there is a representation of the symmetric group associated to G. Also associated to G is a subvariety of the type A flag variety, called a regular semisimple Hessenberg variety. As described by Julianna Tymoczko, there is a nice representation of the symmetric group on the cohomology of this variety. Wachs and I conjectured that, after a sign twist, the two representations just described are the same. This was proved by Patrick Brosnan and Tim Chow. I will discuss this story and some recent developments.
Note: You can find links to relevant background material below.