Kromatic quasisymmetric functions

Abstract

We provide a construction for the kromatic symmetric function ${\overline{X}}_G$ of a graph introduced by Crew, Pechenik, and Spirkl using combinatorial (linearly compact) Hopf algebras. As an application, we show that ${\overline{X}}_G$ has a positive expansion into multifundamental quasisymmetric functions. We also study two related quasisymmetric $q$-analogues of ${\overline{X}}_G$, which are $K$-theoretic generalizations of the quasisymmetric chromatic function of Shareshian and Wachs. We classify exactly when one of these analogues is symmetric. For the other, we derive a positive expansion into symmetric Grothendieck functions when $G$ is the incomparability graph of a natural unit interval order.

38 pages, 2 figures, 3 tables; v2: fixed typos, updated references, added some tables of examples; v3: some corrections and added exposition; v4: updated references, final version