On Cohen-Macaulay Hopf Monoids in Species

In this paper, by Jacob White, the focus is on the study of Cohen-Macaulay Hopf monoids in the category of species. Some insights from the abstract: “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.”

This is interesting for me for several reasons. In a new project I’m working on with Anton Mellit, we are categorifying a formula which arises from Möbius inversion, but which also arises from combinatorial species. In particular, in the theory of species one is able to interpret the plethystic exponentiation of a generating function as a sort of composition of species. One goal in our work is to understand the theory of species at a categorified (homological) level. I am curious if this paper provides some insights into what we are looking for.