On the non-uniqueness of \(q\)-cones of matroids
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.
scientific article; zbMATH DE number 1453335