Hopf monoids of ordered simplicial complexes

TL;DR

This paper explores the Hopf-theoretic structure of pure ordered simplicial complexes, including matroids and related classes, and computes antipodes using topological methods, revealing new algebraic and combinatorial insights.

Contribution

It introduces the concept of Hopf classes for ordered simplicial complexes, systematically studies their properties, and computes antipodes for specific classes like facet-initial complexes and unbounded ordered matroids.

Findings

Defined Hopf classes for ordered simplicial complexes.

Computed antipodes for facet-initial complexes and unbounded ordered matroids.

Embedded the Hopf monoid of ordered matroids into that of ordered generalized permutohedra.

Abstract

We study pure ordered simplicial complexes (i.e., simplicial complexes with a linear order on their ground sets) from the Hopf-theoretic point of view. We define a \textit{Hopf class} to be a family of pure ordered simplicial complexes that give rise to a Hopf monoid under join and deletion/contraction. The prototypical Hopf class is the family of ordered matroids. The idea of a Hopf class allows us to give a systematic study of simplicial complexes related to matroids, including shifted complexes, broken-circuit complexes, and \textit{unbounded matroids} (which arise from unbounded generalized permutohedra with 0/1 coordinates). We compute the antipodes in two cases: \textit{facet-initial complexes} (a much larger class than shifted complexes) and unbounded ordered matroids. In the latter case, we embed the Hopf monoid of ordered matroids into the Hopf monoid of ordered generalized permutohedra, enabling us to compute the antipode using the topological method of Aguiar and Ardila. The calculation is complicated by the appearance of certain auxiliary simplicial complexes that we call \textit{Scrope complexes}, whose Euler characteristics control certain coefficients of the antipode. The resulting antipode formula is multiplicity-free and cancellation-free.