In ongoing joint work with M. Wachs, we study relations between certain symmetric functions, cohomology rings of certain varieties, and permutation statistics. In this talk, I will concentrate on relations between the first two subjects. All terms mentioned below will be defined in the talk.

The chromatic symmetric function of a (finite, loopless) graph G, which was introduced by R. Stanley, encodes at least as much information about G as the well studied chromatic polynomial. It is a symmetric function with integer coefficients in infinitely many variables. We study a refinement of the chromatic symmetric function involving the extension the coefficient ring to a polynomial ring in one variable. In general, our refinement is not symmetric in the original variables. However, for a certain class of graphs our refinement remains symmetric. Through the Frobenius characteristic map, we can associate to a graph G in this class a representation of a finite symmetric group. Also naturally associated to G is a subvariety of the flag variety, called a regular semisimple Hessenberg variety. The theory of torus actions due to M. Goresky-R. Kottwitz-R. MacPherson allows one to define a representation of the same symmetric group on the cohomology ring of this variety. We conjecture that the two representations associated to G are in fact the same. I will present evidence for this conjecture and discuss some actual and potential consequences of its truth.