Flag matroids with coefficients

TL;DR

This paper extends the theory of matroids to flag matroids over arbitrary tracts, introduces their moduli space using ordered blue schemes, and connects these concepts to various geometric contexts including tropical geometry and flag varieties.

Contribution

It generalizes Baker-Bowler theory to flag matroids over any tract, providing new axiomatic descriptions, duality, minors, and a geometric moduli space construction.

Findings

Established duality and minors for flag matroids.

Constructed moduli space of flag matroid bundles.

Connected flag matroids to flag varieties and tropical geometry.

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-Pl\"ucker 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.