Totally free expansions of matroids.
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.
scientific article; zbMATH DE number 1843806
scientific article