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

TL;DR

This paper introduces the Kromatic symmetric function, a K-theoretic analogue of the chromatic symmetric function, exploring its properties, positivity in certain cases, and its relation to existing conjectures and graph invariants.

Contribution

It defines the Kromatic symmetric function, proves positivity results for claw-free incomparability graphs, and shows limitations regarding the Stanley-Stembridge conjecture in K-theory.

Findings

Kromatic symmetric function generalizes chromatic symmetric function to K-theory.

Positivity holds for claw-free incomparability graphs.

Counterexamples show the Stanley-Stembridge conjecture does not lift to K-theory.

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 $X_G$.