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Matroid Theory$
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James Oxley

Print publication date: 2011

Print ISBN-13: 9780198566946

Published to Oxford Scholarship Online: December 2013

DOI: 10.1093/acprof:oso/9780198566946.001.0001

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Representable Matroids

Representable Matroids

(p.158) 6 Representable Matroids
Matroid Theory

James Oxley

Oxford University Press

This chapter provides an overview of the basic questions associated with matroid representability and indicates how one actually goes about constructing representations. The key ideas are presented in Sections 6.1 and 6.3–6.6, which cover projective geometries, different matroid representations, constructing representations for matroids, representability over finite fields, and regular matroids, respectively. Section 6.2 looks at affine geometries, a class of highly symmetric structures that are closely linked to the projective geometries of Section 6.1. Section 6.7 discusses algebraic matroids, a class of matroids that properly contains the class of representable matroids and arises from algebraic dependence over a field. Section 6.8 focuses on characteristic sets, its main idea being concerned with how one can capture geometrically certain algebraic properties of a field. Section 6.9 examines modularity, a special property of flats that is important in several contexts including matroid constructions. Finally, Section 6.10 discusses an important class of matroids introduced by Dowling.

Keywords:   matroid representability, projective geometries, algebraic matroids, characteristic sets, modularity, Dowling

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