Coloring Complexes and Combinatorial Hopf Monoids

TL;DR

This paper extends the concept of coloring complexes to linearized combinatorial Hopf monoids, establishing conditions for their existence, exploring inequalities of associated invariants, and identifying a universal structure within this framework.

Contribution

It introduces a generalized coloring complex for linearized combinatorial Hopf monoids and characterizes when such complexes exist, also identifying a terminal Hopf monoid with convex characters.

Findings

Coloring complexes form a linearized combinatorial Hopf monoid.

Identifies inequalities satisfied by quasisymmetric function invariants.

The collection of all such coloring complexes is the terminal object in a specific category.

Abstract

We generalize the notion of a coloring complex of a graph to linearized combinatorial Hopf monoids. We determine when a linearized combinatorial Hopf monoid has such a construction, and discover some inequalities that are satisfied by the quasisymmetric function invariants associated to the combinatorial Hopf monoid. We show that the collection of all such coloring complexes forms a linearized combinatorial Hopf monoid, which is the terminal object in the category of combinatorial Hopf monoids with convex characters. We also study several examples of combinatorial Hopf monoids.