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Matroid schemes and geometric posets

Christin Bibby

Abstract

The intersection data of a hyperplane arrangement is described by a geometric lattice, or equivalently a simple matroid. There is a rich interplay between this combinatorial structure and the topology of the arrangement complement. In this paper, we characterize the combinatorial structure underlying an abelian arrangement (such as a toric or elliptic arrangement) by defining a class of geometric posets and a generalization of matroids called matroid schemes. The intersection data of an abelian arrangement is encoded in a geometric poset, and we prove that a geometric poset is equivalent to a simple matroid scheme. We lay foundations for the theory of matroid schemes, discussing rank, flats, and independence. We also extend the definition of the Tutte polynomial to this setting and prove that it satisfies a deletion-contraction recurrence.

Matroid schemes and geometric posets

Abstract

The intersection data of a hyperplane arrangement is described by a geometric lattice, or equivalently a simple matroid. There is a rich interplay between this combinatorial structure and the topology of the arrangement complement. In this paper, we characterize the combinatorial structure underlying an abelian arrangement (such as a toric or elliptic arrangement) by defining a class of geometric posets and a generalization of matroids called matroid schemes. The intersection data of an abelian arrangement is encoded in a geometric poset, and we prove that a geometric poset is equivalent to a simple matroid scheme. We lay foundations for the theory of matroid schemes, discussing rank, flats, and independence. We also extend the definition of the Tutte polynomial to this setting and prove that it satisfies a deletion-contraction recurrence.
Paper Structure (16 sections, 26 theorems, 34 equations, 7 figures)

This paper contains 16 sections, 26 theorems, 34 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Matroids and semimatroids
  3. Poset terminology and notation
  4. Matroid schemes
  5. Closure and flats
  6. Independence
  7. Bases
  8. Circuits
  9. Loops and isthmuses
  10. Deletion and contraction
  11. Geometric posets
  12. Tutte polynomial
  13. Characteristic polynomial
  14. Semimatroids with group actions
  15. Abelian arrangements
  16. ...and 1 more sections

Key Result

Proposition 4.5

Let $\mathscr{M}=(S,\rho)$ be a matroid prescheme. Now assume that $\mathscr{M}=(S,\rho)$ is a matroid scheme.

Figures (7)

  • Figure 1: The combinatorial structures underlying arrangements.
  • Figure 2: Depicted are Hasse diagrams for simplicial posets, and each element $x$ labelled by $\rho(x)$. The first two are matroid schemes; the third is a matroid prescheme; the fourth is neither. See \ref{['ex:cw']}.
  • Figure 3: The posets of flats for the matroid schemes depicted in \ref{['fig:ex1', 'fig:ex2', 'fig:nox5']}.
  • Figure 4: The independence poset, bases, and circuits for the matroid scheme $\mathscr{M}$ from \ref{['fig:ex1']}.
  • Figure 5: A matroid scheme $\mathscr{M}$ alongside its contraction by atom $a$ and deletion of atom $a$. See \ref{['ex:contr']}.
  • ...and 2 more figures

Theorems & Definitions (94)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 4.1
  • Remark 4.2
  • Definition 4.3
  • Example 4.4
  • Proposition 4.5
  • proof
  • Definition 4.6
  • ...and 84 more