Stabilizers of Classes of Representable Matroids

Abstract

Let M be a class of matroids representable over a field F . A matroid N ∈ M stabilizes M if, for any 3-connected matroid M ∈ M , an F -representation of M is uniquely determined by a representation of any one of its N -minors. One of the main theorems of this paper proves that if M is minor-closed and closed under duals, and N is 3-connected, then to show that N is a stabilizer it suffices to check 3-connected matroids in M that are single-element extensions or coextensions of N , or are obtained by a single-element extension followed by a single-element coextension. This result is used to prove that a 3-connected quaternary matroid with no U 3, 6 -minor has at most ( q −2)( q −3) inequivalent representations over the finite field GF ( q ). New proofs of theorems bounding the number of inequivalent representations of certain classes of matroids are given. The theorem on stabilizers is a consequence of results on 3-connected matroids. It is shown that if N is a 3-connected minor of the 3-connected matroid M , and | E ( M )− E ( N )|⩾3, then either there is a pair of elements x , y ∈ E ( M ) such that the simplifications of M / x , M / y , and M / x , y are all 3-connected with N -minors or the cosimplifications of M \ x , M \ y , and M \ x , y are all 3-connected with N -minors, or it is possible to perform a Δ − Y or Y − Δ exchange to obtain a matroid with one of the above properties.