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Positroid envelopes and graphic positroids

Jeremy Quail, Puck Rombach

TL;DR

This work establishes a deep link between positroids and graphic matroids by showing that every positroid envelope class contains a graphic matroid and that a graphic positroid is the unique matroid in its envelope class. It develops constructive graph methods to realize positroids and leverages Grassmann necklaces and decorated permutations to organize envelope classes and their decompositions via $2$-sums and series-parallel connections. The authors further relate graphic positroids to incidence-matrix realizations with all nonnegative maximal minors, unifying combinatorial, geometric, and linear-algebraic perspectives. The results yield a comprehensive framework for identifying graphic representatives within positroid envelopes and provide tools for explicit realizations in the totally nonnegative Grassmannian context.

Abstract

Positroids are matroids realizable by real matrices with all nonnegative maximal minors. They partition the ordered matroids into equivalence classes, called positroid envelope classes, by their Grassmann necklaces. We give an explicit graph construction that shows that every positroid envelope class contains a graphic matroid. We prove that a graphic positroid is the unique matroid in its positroid envelope class. Finally, we show that every graphic positroid has an oriented graph representable by a signed incidence matrix with all nonnegative minors.

Positroid envelopes and graphic positroids

TL;DR

This work establishes a deep link between positroids and graphic matroids by showing that every positroid envelope class contains a graphic matroid and that a graphic positroid is the unique matroid in its envelope class. It develops constructive graph methods to realize positroids and leverages Grassmann necklaces and decorated permutations to organize envelope classes and their decompositions via -sums and series-parallel connections. The authors further relate graphic positroids to incidence-matrix realizations with all nonnegative maximal minors, unifying combinatorial, geometric, and linear-algebraic perspectives. The results yield a comprehensive framework for identifying graphic representatives within positroid envelopes and provide tools for explicit realizations in the totally nonnegative Grassmannian context.

Abstract

Positroids are matroids realizable by real matrices with all nonnegative maximal minors. They partition the ordered matroids into equivalence classes, called positroid envelope classes, by their Grassmann necklaces. We give an explicit graph construction that shows that every positroid envelope class contains a graphic matroid. We prove that a graphic positroid is the unique matroid in its positroid envelope class. Finally, we show that every graphic positroid has an oriented graph representable by a signed incidence matrix with all nonnegative minors.
Paper Structure (19 sections, 48 theorems, 73 equations, 2 figures)

This paper contains 19 sections, 48 theorems, 73 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Basic tools and definitions
  3. Matroids
  4. Connectivity, direct sums and $2$-sums
  5. Tree decompositions
  6. Series-parallel matroids
  7. Weak maps
  8. Positroids
  9. Dihedral action
  10. Series-parallel connections and 2-sums of positroids
  11. Related combinatorial objects
  12. Decorated permutations
  13. Grassmann necklaces
  14. Positroid envelopes, envelope classes, and varieties
  15. Positroid envelopes
  16. ...and 4 more sections

Key Result

Theorem 2.1

A matroid is binary if and only if it is $U^2_4$-free.

Figures (2)

  • Figure 1: For a totally ordered set $E$, this diagram commutes.
  • Figure 2: (a) Graph $G$, with $\pi_{M(G)} = (8,3,2,5,4,6,7,1)$. (b) Redrawing of $G$. (c) Graph $G^*$, with $\pi_{(M(G))^*} = (\pi_{M(G)})^{-1}$. (d) Redrawing of $G^*$.

Theorems & Definitions (84)

  • Theorem 2.1: Theorem 6.5.4 in oxley2006matroid
  • Proposition 2.2: Proposition 7.1.21 in oxley2006matroid
  • Definition 2.3: oxley2006matroid
  • Definition 2.4: oxley2006matroid
  • Theorem 2.5: Theorem 8.3.10 in oxley2006matroid
  • Proposition 2.6
  • proof
  • Definition 2.7
  • Corollary 2.8: Corollary 4.7 in borovik1997coxeter
  • Corollary 2.9
  • ...and 74 more