On Perturbations of Highly Connected Dyadic Matroids

Abstract

Geelen, Gerards, and Whittle [3] announced the following result: let q = p k be a prime power, and let M be a proper minor-closed class of GF(q)-representable matroids, which does not contain PG(r − 1, p) for sufficiently high r. There exist integers k, t such that every vertically k-connected matroid in M is a rank-(≤ t) perturbation of a frame matroid or the dual of a frame matroid over GF(q). They further announced a characterization of the perturbations through the introduction of subfield templates and frame templates.We show a family of dyadic matroids that form a counterexample to this result. We offer several weaker conjectures to replace the ones in [3], discuss consequences for some published papers, and discuss the impact of these new conjectures on the structure of frame templates. 1 arXiv:1712.07702v3 [math.CO]In this paper we present a counterexample to this conjecture. In particular, we build a family of dyadic matroids that are vertically kconnected for any desired k, and not a bounded-rank perturbation of either a represented frame matroid or the dual of a represented frame matroid. The construction starts with a cyclically k-connected graph G, modifies it at a number of vertices that grows with |V (G)|, and dualizes the resulting matroid. We detail the construction, and prove its key properties, in Section 3.Vertical connectivity and cographic matroids are not very compatible notions. Because of this, our examples, which are very sparse, arise only in situations where one might expect the second outcome of the conjecture to hold. For this reason, the forthcoming proof of the Matroid Structure Theorem itself is not jeopardized, and versions of Conjecture 1.2 can be recovered. In Section 4, we provide several such conjectures. Included in Section 4 is Section 4.3, where we discuss consequences for [10] and [15]. In Section 5, we discuss consequences for the notion of frame templates, introduced in [3] to describe the perturbations in more detail. PreliminariesUnexplained notation and terminology will generally follow Oxley [16]. One exception is that we denote the vector matroid of a matrix A by M (A), rather than M [A]. The following characterization of the dyadic matroids was shown by Whittle in [23].Theorem 2.1. A matroid is dyadic if and only if it is representable over both GF(3) and GF(5).We will need some definitions and results related to bounded-rank perturbations of represented matroids. The next three definitions are from [3].Definition 2.2. Let M 1 and M 2 be F-represented matroids on a common ground set. Then M 2 is a rank-(≤ t) perturbation of M 1 if there exist matrices A 1 and P such that M (A 1 ) = M 1 , the rank of P is at most t, and M (A 1 + P ) = M 2 . Definition 2.3. Let M 1 and M 2 be F-represented matroids on ground set E. If there is some F-represented matroid M on ground set E ∪ {e} such that M 1 = M \e and M 2 = M/e, then M 1 is an elementary lift of M 2 , and M 2 is an elementary projection of M 1 . Note that an elementary lift of a represented matroid M (A) can be ob...

On the Highly Connected Dyadic, Near-Regular, and Sixth-Root-of-Unity Matroids

“…(We refer the reader to [3] or [15] for the definitions of these matroids.) Hall, Mayhew, and Van Zwam [10, Theorem 1.2], based on unpublished work by Geelen, showed that the excluded minors for the class of near-regular matroids are U 2,5 , U √ 1-matroid of rank r and the largest simple near-regular matroid of rank r. We remark without proof that our main results here, combined with [9,Lemmas 4.14,4.16], show that Hypothesis 3.2 agrees with these known results.…”

“…The notion of frame templates was introduced by Geelen, Gerards, and Whittle in [5] to describe the structure of the highly connected members of minor-closed classes of matroids representable over a fixed finite field. Frame templates have been studied further in [8,14,9,7]. In this section, we give several results proved in those papers that we will need to prove the main results in this paper.…”

“…As stated in Section 1, they are modified versions of a hypothesis given by Geelen, Gerards, and Whittle in [5], and their proofs are forthcoming. In their current forms, these hypotheses were stated in [9]. Hypothesis 3.1 ([9, Hypothesis 4.3]).…”

“…The following definition and theorem are found in [9]. A few specific templates have been given names.…”

“…The results announced in [5] rely on the Matroid Structure Theorem by these same authors [4]. We refer the reader to [9] for more details. Some proofs in this paper involved case checks aided by Version 8.9 of the SageMath software system [18], in particular making use of the matroids component [17].…”