The Templates for Some Classes of Quaternary Matroids

Abstract

Subject to hypotheses based on the matroid structure theory of Geelen, Gerards, and Whittle, we completely characterize the highly connected members of the class of golden-mean matroids and several other closely related classes of quaternary matroids. This leads to a determination of the eventual extremal functions for these classes. One of the main tools for obtaining these results is the notion of a frame template. Consequently, we also study frame templates in significant depth.

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

“…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.…”

“…If Φ and Φ are frame templates, it is possible that M(Φ) = M(Φ ) even though Φ and Φ look very different. There are other notions of template equivalence (namely equivalence, algebraic equivalence, and semi-strong equivalence) given in [7], but all of these imply minor equivalence.…”

“…Frame templates where the groups Γ, Λ, and ∆ are trivial are studied extensively in [7]. Definition 3.9 ([7, Definitions 6.9-6.10]).…”

“…In all of the Y -templates studied in Sections 4 and 5, the matrix P 1 is an empty matrix. However, we use Definition 3.9 in order to stay consistent with [7].…”

“…In Section 2, we give some background information about the classes of matroids studied in this paper. In Section 3, we recall results from [7] that will be used to prove our main results. In Section 4, we prove Theorems 1.1 and 1.2, and in Section 5, we prove Theorem 1.3.…”