Chromatic Quasisymmetric Class Functions for combinatorial Hopf monoids

We study the chromatic quasisymmetric class function of a linearized combinatorial Hopf monoid. Given a linearized combinatorial Hopf monoid $H$, and an $H$-structure $h$ on a set $N$, there are proper colorings of $h$, generalizing graph colorings and poset partitions. We show that the automorphism group of $h$ acts on the set of proper colorings. The chromatic quasisymmetric class function enumerates the fixed points of this action, weighting each coloring with a monomial. For the Hopf monoid of graphs this invariant generalizes Stanley's chromatic symmetric function and specializes to the orbital chromatic polynomial of Cameron and Kayibi. We also introduce the flag quasisymmetric class function of a balanced relative simplicial complex equipped with a group action. We show that, under certain conditions, the chromatic quasisymmetric class function of $h$ is the flag quasisymmetric class function of a balanced relative simplicial complex that we call the coloring complex of $h$. We use this result to deduce various inequalities for the associated orbital polynomial invariants. We apply these results to several examples related to enumerating graph colorings, poset partitions, generic functions on matroids or generalized permutohedra, and others.