On inequivalent representations of matroids over finite fields
type: Item
instance of: scholarly article
title: On inequivalent representations of matroids over finite fields
zbMATH Open document ID: 0856.05021
author: Dirk Vertigan
author: James Oxley
published in: Journal of Combinatorial Theory. Series B
publication date: 1997-02-17T00:00:00Z
review text: It was conjectured by \textit{J. Kahn} [On the uniqueness of matroid representations over GF(4), Bull. Lond. Math. Soc. 20, No. 1, 5-10 (1988; Zbl 0609.05028)] that, for each prime power \(q\), there is an integer \(n(q)\) such that no 3--connected \(\text{GF} (q)\)-representable matroid has more than \(n(q)\) inequivalent \(\text{GF} (q)\)-representations. At the time, this conjecture was known to be true for \(q = 2\) and \(q = 3\), and Kahn had just proved it for \(q = 4\). This paper proves the conjecture for \(q = 5\) showing that 6 is a sharp value for \(n(5)\). Moreover, it is also shown that the conjecture is false for all larger values of \(q\).
Mathematics Subject Classification ID: 05B35
zbMATH DE Number: 934819
zbMATH Keywords: inequivalent representations
zbMATH Keywords: matroid roundedness
zbMATH Keywords: matroid representations
reviewed by: James Oxley
MaRDI profile type: Publication
OpenAlex ID: W2100202683
DOI: 10.1006/JCTB.1996.0049
description: scientific article; zbMATH DE number 934819