On Totally Free Expansions of Matroids

The aim of this paper is to give insight into the behaviour of inequivalent representations of 3{connected matroids. An element x of a matroid M is xed if there is no extension M 0 of M by an element x 0 such that fx;x 0 g is independent and M 0 is unaltered by swapping the labels on x and x 0. When x is xed, a representation of M nx extends in at most one way to a representation of M. A 3{connected matroid N is totally free if neither N nor its dual has a xed element whose deletion is a series extension of a 3{connected matroid. The signiicance of such matroids derives from the theorem, established here, that the number of inequivalent representations of a 3{connected matroid M over a nite eld F is bounded above by the maximum, over all totally free minors N of M , of the number of inequivalent F{representations of N. It is proved that, within a class of matroids that is closed under minors and duality, the totally free matroids can be found by an inductive search. Such a search is employed to show that, for all r 4, there are unique and easily described rank-r quaternary and quinternary matroids, the rst being the free spike. Finally, Seymour's Splitter Theorem is extended by showing that the sequence of 3{connected matroids from a matroid M to a minor N , whose existence is guaranteed by the theorem, may be chosen so that all deletions and contractions of xed and cooxed elements occur in the initial segment of the sequence. 1. Introduction It is by now a truism to say that the presence of inequivalent representations of matroids over elds is the major barrier to progress in matroid representation theory. Strong results giving characterizations of classes of representable matroids certainly in all cases, the class either has a unique representation property, as is the case for binary matroids, ternary matroids over GF(3), or the precise way in which inequivalent representations arise is understood, as is the case for representations of ternary matroids over elds other than GF(3), and qua-ternary matroids over GF(4). What is needed for progress are techniques that would enable one to characterize the way inequivalent representations arise for more general classes of representable matroids. It is with this problem in mind that the research for this paper was undertaken. What …