Séminaire Lotharingien de Combinatoire, 84B.84 (2020), 12 pp.
Jacob A. White
On Cohen-Macaulay Hopf Monoids in Species
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.
Received: November 20, 2019.
Accepted: February 20, 2020.
Final version: April 30, 2020.
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