Stabilizers of classes of representable matroids

Let M be a class of matroids representable over a field F. A matroid N is an element of M stabilizes M if, for any 3-connected matroid M is an element of 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,U-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)\ greater than or equal to 3, then either there is a pair of elements I, y is an element of 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 Delta - Y or Y-Delta exchange to obtain a matroid with one of the above properties. (C) 1999 Academic Press.