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Extension-lifting Bijections for Oriented Matroids

Spencer Backman, Francisco Santos, Chi Ho Yuen

TL;DR

A family of bijections between bases of an oriented matroid and special orientations is described, specified by a pair of circuit and cocircuit signatures coming respectively from a generic single-element lifting and extension.

Abstract

Extending the notion of geometric bijections for regular matroids, introduced by the first and third author with Matthew Baker, we describe a family of bijections between bases of an oriented matroid and special orientations. These bijections are specified by a pair of circuit and cocircuit signatures coming respectively from a generic single-element lifting and extension. We then characterize generic single-element liftings and extensions using these bijections. We also explain the relation of our work with the works of Gioan--Las Vergnas and Ding. Some implications in oriented matroid programming and oriented matroid triangulations are also discussed.

Extension-lifting Bijections for Oriented Matroids

TL;DR

A family of bijections between bases of an oriented matroid and special orientations is described, specified by a pair of circuit and cocircuit signatures coming respectively from a generic single-element lifting and extension.

Abstract

Extending the notion of geometric bijections for regular matroids, introduced by the first and third author with Matthew Baker, we describe a family of bijections between bases of an oriented matroid and special orientations. These bijections are specified by a pair of circuit and cocircuit signatures coming respectively from a generic single-element lifting and extension. We then characterize generic single-element liftings and extensions using these bijections. We also explain the relation of our work with the works of Gioan--Las Vergnas and Ding. Some implications in oriented matroid programming and oriented matroid triangulations are also discussed.

Paper Structure

This paper contains 8 sections, 22 theorems, 4 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results and idea of proof
  3. Connections with Other Works
  4. Background and Organization of the Paper
  5. Preliminaries
  6. Proof of the Main Results
  7. Relation with Orientation Activity and Active Bijections
  8. Relation to Triangulations of Lawrence Polytopes

Key Result

Theorem 1

Let $M$ be an oriented matroid, and let $\sigma$ and $\sigma^*$ be the generic circuit and cocircuit signatures induced from a generic single-element lifting and extension, respectively. Then the map $\beta_{\sigma,\sigma^*}:B \mapsto {\mathcal{O}}(B)$ is a bijection between the set of bases of $M$

Figures (1)

  • Figure 1: Left: The affine pseudohyperplane arrangement of $M(K_3)$ (the three curves represent the elements of $M$), together with the extra elements $g$. The regions are labeled by $\sigma$-compatible orientations of $M$. Right: The new curve represents $f$. There are three regions whose optima with respect to $f$ are bounded, and each of these optima is the intersection of curves that form a basis of $M$.

Theorems & Definitions (48)

  • Definition 1.1
  • Theorem 1
  • Theorem 2
  • Remark 1.2
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3: Las Vergnas, see BLSWZ_book
  • Definition 2.4
  • Theorem 2.5
  • Lemma 2.6: Las Vergnas
  • ...and 38 more