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Flag matroids with coefficients

Manoel Jarra, Oliver Lorscheid

Abstract

This paper is a direct generalization of Baker-Bowler theory to flag matroids, including its moduli interpretation as developed by Baker and the second author for matroids. More explicitly, we extend the notion of flag matroids to flag matroids over any tract, provide cryptomorphic descriptions in terms of basis axioms (Grassmann-Plücker functions), circuit/vector axioms and dual pairs, including additional characterizations in the case of perfect tracts. We establish duality of flag matroids and construct minors. Based on the theory of ordered blue schemes, we introduce flag matroid bundles and construct their moduli space, which leads to algebro-geometric descriptions of duality and minors. Taking rational points recovers flag varieties in several geometric contexts: over (topological) fields, in tropical geometry, and as a generalization of the MacPhersonian.

Flag matroids with coefficients

Abstract

This paper is a direct generalization of Baker-Bowler theory to flag matroids, including its moduli interpretation as developed by Baker and the second author for matroids. More explicitly, we extend the notion of flag matroids to flag matroids over any tract, provide cryptomorphic descriptions in terms of basis axioms (Grassmann-Plücker functions), circuit/vector axioms and dual pairs, including additional characterizations in the case of perfect tracts. We establish duality of flag matroids and construct minors. Based on the theory of ordered blue schemes, we introduce flag matroid bundles and construct their moduli space, which leads to algebro-geometric descriptions of duality and minors. Taking rational points recovers flag varieties in several geometric contexts: over (topological) fields, in tropical geometry, and as a generalization of the MacPhersonian.
Paper Structure (38 sections, 30 theorems, 107 equations)

This paper contains 38 sections, 30 theorems, 107 equations.

Table of Contents

  1. Baker-Bowler theory
  2. Tracts
  3. Matroids over tracts
  4. Push-forwards and the underlying matroid
  5. Circuits
  6. Duality
  7. Orthogonality
  8. Dual pairs
  9. Minors
  10. Examples
  11. Flag matroids
  12. Definitions
  13. Functoriality
  14. Cryptomorphism
  15. Flag matroids as flag K-matroids
  16. ...and 23 more sections

Key Result

Proposition 1

Let ${\mathbf M}=(M_1,\dotsc,M_s)$ be a sequence of matroids $M_i$ with respective Grassmann-Plücker functions $\myvarphi_i:E^{r_i}\to {\mathbb K}$. Then ${\mathbf M}$ is a flag matroid if and only if for all $1\leqslant i\leqslant j\leqslant s$ and $x_1,\dotsc,x_{r_i-1},y_1,\dotsc,y_{r_j+1}\in E$, is in the nullset $N_{\mathbb K}$ where $\myepsilon=1$ in this case.

Theorems & Definitions (83)

  • Example : Flags of matroid minors
  • Proposition 1
  • Definition
  • Theorem A
  • Theorem B
  • Example : Flags of linear subspaces as flag $K$-matroids
  • Example : Tropical flag matroids as flag ${\mathbb T}$-matroids
  • Example : Flag ${{\mathbb F}_1^\pm}$-matroids
  • Example : Oriented flag matroids
  • Theorem C
  • ...and 73 more