Representation of matroids with a modular plane

2013/04/24 by Jim Geelen, Geelen, Jim, Rohan Kapadia +1
Mathematics · #05B35 #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:05B35

paper · pdf · doi:10.48550/arxiv.1304.6448

59 pages. Submitted

arxiv created 2013/04/24 · arxiv updated 2013/04/25

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, q²+1}-minor. We also show that no excluded minor for the class of F-representable matroids has a PG(2, F)-restriction.

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