Representation of matroids with a modular plane

TL;DR

This paper establishes conditions under which matroid representations extend uniquely and characterizes the structure of certain matroids with modular flats, impacting understanding of matroid representability over finite fields.

Contribution

It proves that representations of matroids with a modular flat extend uniquely and characterizes matroids with PG(2, F)-restrictions, linking representability and minors.

Findings

Unique extension of representations for matroids with modular flats.

Characterization of matroids with PG(2, F)-restrictions as either F-representable or having a specific minor.

No excluded minor for F-representability contains a PG(2, F)-restriction.

Abstract

We prove that if M is a vertically 4-connected matroid with a modular flat X of rank at least three, then every representation of M | X over a finite field F extends to a unique F-representation of M. A corollary is that when F has order q, any vertically 4-connected matroid with a PG(2, F)-restriction is either F-representable or has a U_{2, q^2+1}-minor. We also show that no excluded minor for the class of F-representable matroids has a PG(2, F)-restriction.