On the non-uniqueness of \(q\)-cones of matroids
type: Item
instance of: scholarly article
title: On the non-uniqueness of \(q\)-cones of matroids
zbMATH Open document ID: 0957.05024
DOI: 10.1016/S0012-365X(99)00358-1
published in: Discrete Mathematics
publication date: 2001-03-08T00:00:00Z
review text: Let \(M\) be a rank-\(r\) simple \(\text{GF}(q)\)-representable matroid. A \(q\)-cone of \(M\) (originally introduced by Whittle as a \(q\)-lift) is a matroid \(M'\) that is constructed by embedding \(M\) in a hyperplane \(H\) of \(\text{PG}(r,q)\), adding a point \(p\) of \(\text{PG}(r,q)\) not on \(H\) and then adding all the points of \(\text{PG}(r,q)\) that are on lines joining \(p\) to an element of \(M\). Kung noted that the \(q\)-cones of two equivalent \(\text{GF}(q)\)-representations of a matroid \(M\) are isomorphic and asked whether this must also be the case for two inequivalent \(\text{GF}(q)\)-representations. In this note it is shown that this is not necessarily the case. In particular, two examples are presented (one with \(M\) rank-2 and the other with \(M\) rank-3) which show that if the rank of \(M\) is 2 or if \(M\) is not uniquely representable over \(\text{GF}(q)\), then \(M'\) need not be unique.
Mathematics Subject Classification ID: 05B35
zbMATH DE Number: 1453335
zbMATH Keywords: \(q\)-cone
zbMATH Keywords: representable matroid
author: James Oxley
author: Geoff Whittle
reviewed by: Vincent Moulton
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1453335