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Hopf monoids of ordered simplicial complexes

Federico Castillo, Jeremy L. Martin, Jose A. Samper

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.

Hopf monoids of ordered simplicial complexes

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.

Paper Structure

This paper contains 29 sections, 37 theorems, 106 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Background: Matroids and combinatorial Hopf theory
  3. Ordered simplicial complexes
  4. Polyhedra
  5. Antipodes
  6. Background and notation
  7. Simplicial complexes
  8. Set compositions, preposets, and the braid fan
  9. Generalized permutohedra
  10. Hopf monoids
  11. Hopf classes of ordered complexes
  12. Definitions and basic properties
  13. A zoo of Hopf classes
  14. Hopf monoids from Hopf classes
  15. Ordered generalized permutohedra
  16. ...and 14 more sections

Key Result

theorem 1.1

(Proposition universal-Hopf + Theorem thm:quasiHopf) Every Hopf class $\textbf{h}$ gives rise to a commutative Hopf monoid $\mathbf{H}$ that is a vector subspecies (though not a Hopf submonoid) of $\mathbf{L}^*\times\mathbf{SC}$. Moreover, the Hopf class $\textbf{pre}$ of prefix-pure ordered complex

Figures (6)

  • Figure 1: The normal fan of the cone in Example \ref{['ex:cone']}.
  • Figure 2: The hierarchy of Hopf classes of prefix-pure complexes. Solid lines indicate inclusions known to be strict; dashed lines indicate possible equalities. All classes are described in §§\ref{['sec:Hopf-class-basic']}--\ref{['sec:Hopf-class-zoo']}, except $\textbf{omat}^+$, which will be described in §\ref{['omatplus-from-hclass']}.
  • Figure 3: An unbounded 0/1-EGP whose indicator complex is not a matroid.
  • Figure 4: A shifted matroid and an interval minor, represented as Ferrers diagrams.
  • Figure 5: Locality for the hypersimplex $\Delta_{2,4}$.
  • ...and 1 more figures

Theorems & Definitions (97)

  • theorem 1.1
  • theorem 1.2
  • theorem 1.3
  • theorem 1.4
  • theorem 1.5
  • conjecture 1.6
  • proposition 2.1
  • lemma 2.2
  • proof
  • example 2.3: A cone
  • ...and 87 more