On Cohen-Macaulay Hopf monoids in species

TL;DR

This paper explores Cohen-Macaulay properties of Hopf monoids in species, linking algebraic invariants to topological structures, and applies these insights to graph colorings and posets.

Contribution

It establishes conditions under which polynomial invariants from Hopf monoids are Hilbert polynomials of relative simplicial complexes, connecting algebraic and topological combinatorics.

Findings

Polynomial invariants can be interpreted as Hilbert polynomials of relative simplicial complexes.

Cohen-Macaulay Hopf monoids yield nonnegative h-vectors in associated complexes.

Results apply to chromatic polynomials of acyclic graphs and order polynomials of double posets.

Abstract

We study Cohen-Macaulay Hopf monoids in the category of species. The goal is to apply techniques from topological combinatorics to the study of polynomial invariants arising from combinatorial Hopf algebras. Given a polynomial invariant arising from a linearized Hopf monoid, we show that under certain conditions it is the Hilbert polynomial of a relative simplicial complex. If the Hopf monoid is Cohen-Macaulay, we give necessary and sufficient conditions for the corresponding relative simplicial complex to be relatively Cohen-Macaulay, which implies that the polynomial has a nonnegative $h$-vector. We apply our results to the weak and strong chromatic polynomials of acyclic mixed graphs, and the order polynomial of a double poset.