Convex Geometries via Hopf Monoids: Combinatorial Invariants, Reciprocity, and Supersolvability

TL;DR

This paper explores the combinatorial invariants of convex geometries within Hopf monoids, providing new reciprocity theorems, combinatorial descriptions, and geometric interpretations of associated polynomials and quasisymmetric functions.

Contribution

It introduces a unified framework for polynomial and quasisymmetric invariants of convex geometries, including reciprocity theorems and supersolvability criteria.

Findings

Proved combinatorial reciprocity theorems for key polynomials.

Connected coefficients of invariants to faces of simplicial complexes.

Provided geometric interpretations of the $ab$- and $cd$-index coefficients.

Abstract

We study the Hopf monoid of convex geometries, which contains partial orders as a Hopf submonoid, and investigate the combinatorial invariants arising from canonical characters. Each invariant consists of a pair: a polynomial and a more general quasisymmetric function. We give combinatorial descriptions of the polynomial invariants and prove combinatorial reciprocity theorems for the Edelman-Jamison and Billera-Hsiao-Provan polynomials, which generalize the order and enriched order polynomials, respectively, within a unified framework. For the quasisymmetric invariants, we show that their coefficients enumerate faces of certain simplicial complexes, including subcomplexes of the Coxeter complex and a simplicial sphere structure introduced by Billera, Hsiao, and Provan. We also examine the associated $ab$- and $cd$-indices. We establish an equivalent condition for convex geometries to be supersolvable and use this result to give a geometric interpretation of the $ab$- and $cd$-index coefficients for this class of convex geometries.