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Flats and hyperplane arrangements for matroids with coefficients

Jannis Koulman, Oliver Lorscheid

Abstract

Based on the notion of vectors and linear subspaces for a matroid, we develop a theory of flats and hyperplane arrangements for T-matroids, where T is a tract. This leads to several cryptomorphic descriptions of T-matroids: in terms of its lattice of T-flats, as a hyperplane arrangement over T, as point-line arrangements in projective space over T and as a quiver representation over T. We examplify these notions in the case of tropical linear spaces, a.k.a. valuated matroids.

Flats and hyperplane arrangements for matroids with coefficients

Abstract

Based on the notion of vectors and linear subspaces for a matroid, we develop a theory of flats and hyperplane arrangements for T-matroids, where T is a tract. This leads to several cryptomorphic descriptions of T-matroids: in terms of its lattice of T-flats, as a hyperplane arrangement over T, as point-line arrangements in projective space over T and as a quiver representation over T. We examplify these notions in the case of tropical linear spaces, a.k.a. valuated matroids.
Paper Structure (22 sections, 27 theorems, 36 equations)

This paper contains 22 sections, 27 theorems, 36 equations.

Table of Contents

  1. Background on Baker-Bowler theory
  2. Tracts
  3. Grassmann-Plücker functions
  4. T-representations of matroids
  5. Duality
  6. Minors
  7. Direct sums
  8. Vectors
  9. Linear subspaces
  10. Matroid quotients
  11. T-flats
  12. F-quotients
  13. Cryptomorphism through F-quotients of ranks 1 and 2
  14. Lattices of T-flats
  15. Point-line arrangements
  16. ...and 7 more sections

Key Result

Theorem A

A collection ${\mathbf V}$ of linear subspaces of $T^n$ is the lattice of $T$-flats of a $T$-matroid $[\eta]$ if and only if it satisfies that

Theorems & Definitions (60)

  • Remark
  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Proposition 1
  • Example 1
  • Remark 1
  • Theorem 1.1
  • Corollary 1
  • ...and 50 more