# Hessenberg varieties and chromatic quasisymmetric functions

A Hessenberg function $h$ associates two objects; one is "an incomparability graph of natural unit interval order $G_h$" and the other is "a regular semisimple Hessenberg variety $X(h)$". Shareshian and Wachs conjecture an equivalence between the chromatic quasisymmetric function of $G_h$ and the representation of the symmetric group on the cohomology of $X(h)$ introduced by Tymoczko. In this talk, I state the conjecture and report a partial affirmative answer to the conjecture. I'm planning to explain the chromatic quasisymmetric function of $G_h$ in the first half and the representation of the symmetric group on the cohomology of $X(h)$ in the latter half. This is a joint work with Hiraku Abe, Megumi Harada and Mikiya Masuda.