Stabilizers of Classes of Representable Matroids

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 U3, 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.