Kromatic quasisymmetric functions

TL;DR

This work recasts the kromatic symmetric function $\overline X_G$ within the framework of linearly compact combinatorial Hopf algebras to reveal its algebraic origins and to develop $K$-theoretic/$q$-deformed refinements. It constructs a natural LC-Hopf algebra approach that extends graphs, weighted graphs, and related combinatorial objects (via set-valued P-partitions and acyclic multi-orientations) to the unbounded-degree setting, producing a positive multifundamental expansion for $\overline X_G$ and two $q$-analogues with a detailed symmetry analysis. The paper also connects these constructions to symmetric Grothendieck functions in special cases (notably incomparability graphs of natural unit interval orders) and provides explicit expansions and tableaux-based interpretations, thereby unifying and extending prior results of CPS and Shareshian–Wachs. The results offer new algebraic tools for studying chromatic-type invariants in graphs and posets, with potential applications to $K$-theory and quasisymmetric function theory.

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.

Figures (2)

• Figure 1: The distinct acyclic multi-orientations with 6 and 7 vertices for the complete graph with vertex set $\{v,w\}$. Here, we have written the vertices of these directed graphs as $v_i$ and $w_i$ instead of $(v,i)$ and $(w,i)$ to save space.
• Figure 2: The directed graph $\tilde{D}$ corresponding to Example \ref{['STD-ex']}, with some edges omitted and with vertices $(v,i)$ and $(w,j)$ written as $v_i$ and $w_j$. The boxed vertices are the elements of $\tilde{S}\setminus \tilde{T}$ while the unboxed vertices are the elements of $\tilde{T}$, and $\tilde{S} \cap \tilde{T} = \{v_3,w_4\}$. This picture does not show all of $\tilde{D}$, which has additional directed edges from each boxed vertex to each unboxed vertex.

Theorems & Definitions (58)

• Definition 1.1: Stanley Stanley95
• Example 1.2
• Definition 1.3: SW2016
• Example 1.4
• Definition 1.5: Crew, Pechenik, and Spirkl CPS
• Example 1.6
• Remark 1.7
• Example 2.1
• Example 2.2
• Example 2.3