Weak Maps and Stabilizers of Classes of Matroids

Abstract

Let F be a field and let N be a matroid in a class N N of F-representable matroids that is closed under minors and the taking of duals. Then N is an F-stabilizer for N N if every representation of a 3-connected member of N N is determined up to elementary row operations and column scaling by a representation of any one of its N-minors. The study of stabilizers was initiated by Whittle. This paper extends that study by examining certain types of stabilizers and considering the connection with weak maps.The notion of a universal stabilizer is introduced to identify the underlying matroid structure that guarantees that N will be an FЈ-stabilizer for N N for every field FЈ over which members of N N are representable. It is shown that, just as with F-stabilizers, one can establish whether or not N is a universal stabilizer for N N by an elementary finite check. If N is a universal stabilizer for N N, we determine additional conditions on N and N N that ensure that if N is not a strict rank-preserv-*E-mail: jfgeelen@jeeves.uwaterloo.ca. † E-mail: oxley@math.lsu.edu. ‡ E-mail: vertigan@math.lsu.edu. § E-mail: whittle@kauri.vuw.ac.nz.

Totally Free Expansions of Matroids

“…In a sense, this paper is the third in a series seeking to develop such techniques, the others being [10,27]. The motivation for this paper was that a promising idea, developed in [10], did not turn out to be quite as fruitful as we had hoped. In that paper the notion of a ''universal stabilizer'' for a well-closed class of matroids was introduced.…”

Generalized Δ–Y Exchange and k-Regular Matroids

“…The first of these is easily seen to be implicit in the first paragraph of the proof of Theorem 5.1 of [8].…”

“…This contradiction completes the proof of the lemma. K As noted in [8], it is immediate from the definition of clones that elements x and x$ are clones in M if and only if they are clones in M*. Also, recall from the last section that if x and x$ are clones of a matroid M, and N is a minor of M containing [x, x$], then x and x$ are clones in N. We shall use both these facts in the next result, the first of two corollaries of the last lemma.…”

“…Let N be a member of a well-closed class of matroids N. The notion of a universal stabilizer was introduced in [8] to identify the underlying matroid structure that ensures that, whenever P is a partial field over which N is representable, N is a P-stabilizer for all members of N which are P-representable. Indeed, we have the following result [8, Theorem 5.1].…”

“…Each of these proofs relies on the theory of``stabilizers'' and``universal stabilizers'' initiated in [30] and [8], respectively. We now outline the definitions and results from these papers that will be used in proving Theorems 1.3 and 1.4.…”