Séminaire Lotharingien de Combinatoire, 89B.77 (2023), 12 pp.
Logan Crew, Oliver Pechenik and Sophie Spirkl
The Kromatic Symmetric Function: A K-Theoretic Analogue of XG
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 their K-analogues, the basis of symmetric Grothendieck functions.
We introduce and initiate a theory of the Kromatic symmetric function
X-G, a K-theoretic analogue of the chromatic symmetric function XG 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,
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 X-G and show that it admits an edge deletion-contraction relation. Finally, we give a K-analogue for X-G of the classic monomial-basis expansion of XG.
Received: November 15, 2022.
Accepted: February 20, 2023.
Final version: April 1, 2023.
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