A characterisation of the matroids representable over GF(3) and the rationals
type: Item
instance of: scholarly article
title: A characterisation of the matroids representable over GF(3) and the rationals
zbMATH Open document ID: 0835.05015
published in: Journal of Combinatorial Theory. Series B
publication date: 1996-04-08T00:00:00Z
review text: It follows from a fundamental (1958) result of Tutte that a binary matroid is representable over the rationals if and only if it can be represented by a totally unimodular matrix, that is, by a matrix over the rationals with the property that all subdeterminants belong to \(\{0,1, - 1\}\). For an arbitrary field \({\mathbf F}\), it is of interest to ask for a matrix characterisation of those matroids representable over \({\mathbf F}\) and the rationals. In this paper this question is answered when \({\mathbf F}\) is GF(3). It is shown that a ternary matroid is representable over the rationals if and only if it can be represented over the rationals by a matrix \(A\) with the property that all subdeterminants of \(A\) belong to the set \(\{0, \pm 2^i : i\) an integer\}. While ternary matroids are uniquely representable over GF(3), this is not generally the case for representations of ternary matroids over other fields. A characterisation is given of the class of ternary matroids that are uniquely representable over the rationals.
Mathematics Subject Classification ID: 05B35
zbMATH DE Number: 839123
zbMATH Keywords: binary matroid
zbMATH Keywords: unimodular matrix
zbMATH Keywords: matrix characterisation
zbMATH Keywords: ternary matroid
zbMATH Keywords: representations
author: Geoff Whittle
Wikidata QID: Q56209797
MaRDI profile type: Publication
full work available at URL: jctb.1995.1052
OpenAlex ID: W1967097051
DOI: 10.1006/JCTB.1995.1052
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description: scientific article; zbMATH DE number 839123