Totally free expansions of matroids.
type: Item
instance of: scholarly article
title: Totally free expansions of matroids.
zbMATH Open document ID: 1048.05020
author: Dirk Vertigan
author: James F. Geelen
author: James Oxley
author: Geoff Whittle
published in: Journal of Combinatorial Theory. Series B
publication date: 2002-12-10T00:00:00Z
review text: An element \(x\) of a matroid \(M\) is fixed if there is no extension \(M'\) of \(M\) by an element \(x'\) such that \(\{ x,x'\}\) is independent and \(x\) and \(x'\) are clones, i.e. \(M'\) is unaltered by swapping the lables on \(x\) and \(x'\). An element has freedom, if it is not fixed. To illustrate the usefulness of the notion of clones the authors provide a simple proof of the fact that if \(M\) is \(\mathbf F\)-representable with \(x\) (co)fixed, then a representation of \(M\setminus x (M/x\)) that (co)extends to a representation of \(M\) does so uniquely. The convers to this proposition is conjectured not to hold. A 3-connected matroid \(N\) is called totally free if neither \(N\) nor \(N^*\) has a fixed element whose deletion is a series extension of a 3-connected matroid. By a careful analysis of the freedom of an element in the matroid and its dual, the authors derive, via a sequence of structural lemmas, a significant application of totally free matroids. The number of inequivalent representations of a 3-connected matroid over a finite field \(\mathbf F\) is bounded above by the maximum possible number of inequivalent \(\mathbf F\)-representations of \(N\), where \(N\) is a totally free minor of \(M\). Moreover, it is shown that the totally free minors can be found inductively.
Mathematics Subject Classification ID: 05B35
zbMATH DE Number: 1843806
zbMATH Keywords: inequivalent representations
zbMATH Keywords: 3-connected matroids
zbMATH Keywords: totally free matroid
zbMATH Keywords: clone
reviewed by: Herman J. Servatius
MaRDI profile type: Publication
OpenAlex ID: W1990357453
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DOI: 10.1006/JCTB.2001.2068
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description: scientific article; zbMATH DE number 1843806