The Kromatic Symmetric Function: A $K$-theoretic Analogue of $X_G$

Abstract

Schur functions are a basis of the symmetric function ring that represent Schubert cohomology classes for Grassmannians. Replacing the cohomology ring with $K$-theory yields a rich combinatorial theory of inhomogeneous deformations, where Schur functions are replaced by their $K$-analogues, the basis of symmetric Grothendieck functions. We introduce and initiate a theory of the Kromatic symmetric function $\overline{X}_G$, a $K$-theoretic analogue of the chromatic symmetric function $X_G$ of a graph $G$. The Kromatic symmetric function is a generating series for graph colorings in which vertices may receive any nonempty set of distinct colors such that neighboring color sets are disjoint. Our main result lifts a theorem of Gasharov (1996) to this setting, showing that when $G$ is a claw-free incomparability graph, $\overline{X}_G$ is a positive sum of symmetric Grothendieck functions. This result suggests a topological interpretation of Gasharov's theorem. We then show that the Kromatic symmetric functions of path graphs are not positive in any of several $K$-analogues of the $e$-basis of symmetric functions, demonstrating that the Stanley-Stembridge conjecture (1993) does not have such a lift to $K$-theory and so is unlikely to be amenable to a topological perspective. We also define a vertex-weighted extension of $\overline{X}_G$ and show that it admits a deletion--contraction relation. Finally, we give a $K$-analogue for $\overline{X}_G$ of the classic monomial-basis expansion of $X_G$.