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 ??Y or Y?? exchange to obtain a matroid with one of the above properties.

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