The excluded minors for $\mathrm{GF}(5)$ -representable matroids on ten elements

Nick Brettell School of Mathematics and Statistics
Victoria University of Wellington
New Zealand nick.brettell@vuw.ac.nz

Abstract.

Mayhew and Royle (2008) showed that there are $564$ excluded minors for the class of $\mathrm{GF}(5)$ -representable matroids having at most $9$ elements. We enumerate the excluded minors for $\mathrm{GF}(5)$ -representable matroids having $10$ elements: there are precisely 2128 such excluded minors. In the process we find, for each $i\in\{2,3,4\}$ , the excluded minors for the class of $\mathbb{H}_{i}$ -representable matroids having at most $10$ elements, and the excluded minors for the class of $\mathbb{H}_{5}$ -representable matroids having at most $13$ elements.

Supported by the ERC consolidator grant 617951, and the New Zealand Marsden Fund.

1. Introduction

The excluded-minor characterisations for binary, ternary, and quaternary matroids are seminal results in matroid theory: there are 1, 4, and 7 excluded minors for these classes, respectively [25, 24, 3, 14]. For the class of matroids representable over $\mathrm{GF}(5)$ , the excluded-minor characterisation is not known. However, in 2008 Mayhew and Royle enumerated all matroids on at most nine elements, and thereby found that there are precisely 564 excluded minors on at most nine elements [16].

These days, one can easily enumerate all 383172 matroids on at most nine elements on their home computer should they wish¹¹1For example, see the MatroidUnion blog post here: http://matroidunion.org/?p=301., but enumerating all $10$ -element matroids still appears well out of reach (Mayhew and Royle estimate there are at least $2.5\times 10^{12}$ sparse paving matroids with rank 5 and $10$ elements [16]). However, here we enumerate all those that are excluded minors for the class of $\mathrm{GF}(5)$ -representable matroids.

Theorem 1.1.

There are $2128$ excluded minors for $\mathrm{GF}(5)$ -representability on $10$ elements.

Our approach to find these excluded minors uses the theory of Hydra- $i$ partial fields developed by Pendavingh and van Zwam [21]. A $3$ -connected matroid representable over $\mathrm{GF}(5)$ has at most six inequivalent representations [20]. Let $M$ be a $3$ -connected matroid with a $U_{2,5}$ - or $U_{3,5}$ -minor. Pendavingh and van Zwam showed that, for each $i\in\{1,2,3,4,5,6\}$ , there is a partial field $\mathbb{H}_{i}$ such that $M$ is $\mathbb{H}_{i}$ -representable if and only if $M$ has at least $i$ inequivalent representations over $\mathrm{GF}(5)$ . In particular, $\mathbb{H}_{1}=\mathrm{GF}(5)$ . Moreover, $\mathbb{H}_{5}=\mathbb{H}_{6}$ ; that is, if $M$ has at least five inequivalent representations over $\mathrm{GF}(5)$ , then it has precisely six. For $i\in\{1,2,3,4\}$ , let $\mathcal{N}_{i}$ be the set of excluded minors for the class of $\mathbb{H}_{i+1}$ -representable matroids that are $\mathbb{H}_{i}$ -representable. Then, the matroids in $\mathcal{N}_{i}$ are strong stabilizers for the class of $\mathbb{H}_{i}$ -representable matroids [26]; loosely speaking, this means that when considering a $3$ -connected matroid $M$ in this class with some matroid $N$ in $\mathcal{N}_{i}$ as a minor, then a representation of $N$ uniquely extends to a representation of $M$ . Thus, using this theory, one can first find the excluded minors for $\mathbb{H}_{5}$ -representability on at most $10$ elements, then let $\mathcal{N}_{4}$ be those that are $\mathbb{H}_{4}$ -representable, and then use these as seeds in order to find the excluded minors for $\mathbb{H}_{4}$ -representability on at most $10$ elements. This process can then be repeated to find the excluded minors for $\mathbb{H}_{3}$ -representability on at most $10$ elements. After two further iterations, one obtains the excluded minors for $\mathbb{H}_{1}$ -representability on at most $10$ elements, and these are the excluded minors for the class of $\mathrm{GF}(5)$ -representable matroids, just under another name.

As a slightly more subtle point, note that when an excluded minor $N$ for the class of $\mathbb{H}_{i}$ -representable matroids is $\mathrm{GF}(5)$ -representable, for some $i\in\{2,3,4,5\}$ , it need not be $\mathbb{H}_{i-1}$ -representable. For example, $F_{7}^{-}$ is an excluded minor for $\mathbb{H}_{5}$ -representable matroids that is $\mathbb{H}_{2}$ -representable, but not $\mathbb{H}_{3}$ - or $\mathbb{H}_{4}$ -representable. In this case, where $N$ is $\mathbb{H}_{j}$ -representable, for some $j<i-1$ , but is not $\mathbb{H}_{j+1}$ -representable, $N$ is an excluded minor for the class of $\mathbb{H}_{\ell}$ -representable matroids for each $\ell\in\{j+1,j+2,\dotsc,i\}$ , and, in particular, it is a member of $\mathcal{N}_{j}$ , so will be used as a seed to find excluded minors for the class of $\mathbb{H}_{j}$ -representable matroids.

The resolution of Rota’s conjecture [12] guarantees that there is a finite number of excluded minors for the class of $\mathrm{GF}(5)$ -representable matroids. There are certainly excluded minors for this class on more than 10 elements; in particular, there is an excluded minor on 12 elements, on 14 elements, and on 16 elements [8]. It is natural to ask the value of finding an incomplete list of excluded minors. For the excluded-minor characterisations of representable matroids that are currently known exactly, the excluded minors have at most 8 elements [14, 15, 7], and there are at most 17 of them in total [7], so there is little that can be gleaned regarding how one might expect the excluded minors of increasing sizes to be distributed. However, in light of Theorem 1.1, it seems likely that so far we have only discovered the tip of the iceberg (whereas one could have been a little bit more optimistic if only aware of those on at most 9 elements). With effort, or more computing power, one might be able to extend the search beyond 10 elements, but reaching (or exceeding) 16 elements appears a long way off. Nonetheless, in the author’s opinion, it is worthwhile to explore how bad things might be, and there is precedent for doing this for minor-closed classes of graphs having similarly unwieldy excluded-minor characterisations [28, 17].

More concretely, a component of this work, when combined with [7], gets us tantalisingly close to a complete list of the excluded-minors for the class of $\mathbb{H}_{5}$ -representable matroids. It turns out that the class of $\mathbb{H}_{5}$ -representable matroids is closely related to another well-studied class. For an integer $k\geq 2$ , the class of $k$ -regular matroids is an algebraic generalisation of the class of regular and near-regular matroids, where regular matroids correspond to $0$ -regular matroids, and near-regular matroids correspond to $1$ -regular matroids (a formal definition of these classes is given in Section 2). Although the class of near-regular matroids coincides with the class of matroids representable over all fields of size at least three [27], the class of $2$ -regular matroids is a proper subclass of matroids representable over all fields of size at least four [23]. However, the class of $3$ -regular matroids coincides with the class of $\mathbb{H}_{5}$ -representable matroids [7]. Thus, attempts to discover the excluded minors for $\mathbb{H}_{5}$ -representable matroids can equivalently be viewed as attempts to discover the excluded minors for $3$ -regular matroids.

Brettell, Oxley, Semple, and Whittle showed that an excluded minor for the class of $2$ -regular matroids, or for the class of $3$ -regular matroids, has at most $15$ elements [7]. Brettell and Pendavingh computed the excluded minors for the class of $2$ -regular matroids of size at most $15$ [8]. Thus, an excluded-minor characterisation for the class of $2$ -regular matroids is now known, and this is an important step towards understanding matroids representable over all fields of size at least four.

To find the excluded minors for $\mathrm{GF}(5)$ -representability of size at most $10$ , using the approach described earlier, the first step is to find the excluded minors for the class of $\mathbb{H}_{5}$ -representable matroids of size at most $10$ . However, to complement the aforementioned known bound on an excluded minor for the class of $\mathbb{H}_{5}$ -representable matroids, we extend our search beyond matroids of size $10$ . We do not quite match the bound of $15$ , however; we find all excluded minors for the class having at most $13$ elements.

Theorem 1.2.

There are exactly $33$ matroids that are excluded minors for the class of $\mathbb{H}_{5}$ -representable matroids and have at most $13$ elements.

Descriptions of these $33$ matroids appear in Section 4 (see Theorem 4.5).

For Theorem 1.1, we borrow from the approach taken in [8]; however, to find the excluded minors for $\mathrm{GF}(5)$ -representability of size at most $10$ , the computational techniques we require are less advanced, since we are only dealing with matroids of size at most $10$ . On the other hand, as we are dealing with much larger sets of excluded minors and stabilizers, careful book-keeping is required. For Theorem 1.2, the more advanced techniques employed in [8] become useful; in particular, when considering matroids of size $n\geq 11$ , we can dramatically speed up the computations by only considering compatible pairs of extensions of matroids of size $n-2$ (rather than all extensions of matroids of size $n-1$ ). For this approach to be exhaustive while only keeping track of $3$ -connected matroids with a certain $N$ -minor, we require a splitter theorem for $N$ -detachable pairs [9]. Moreover, there are particular $12$ -element matroids for which we cannot guarantee the existence of $N$ -detachable pairs; we study these further, to show they are not excluded minors for the class of $\mathbb{H}_{5}$ -representable matroids.

One final observation from this work is that an excluded-minor characterisation for $\mathbb{H}_{4}$ -representable matroids, though extremely difficult, could perhaps also be achievable, putting it in a similar category to dyadic matroids and matroids representable over all fields of size at least four.

This paper is structured as follows. In the next section, we review partial fields, stabilizers, and other related notions that we need to describe the methodology, and argue the correctness, of our approach. In the two subsequent sections, we consider the innermost layer of the Hydra hierarchy, $\mathbb{H}_{5}$ -representable matroids. First, we study $12$ -element matroids that have no $N$ -detachable pairs, in Section 3. Then, in Section 4, we obtain the excluded minors for the class of $\mathbb{H}_{5}$ -representable matroids on at most $13$ elements. In Section 5, we consider the excluded minors for the remaining layers of the Hydra hierarchy: that is, the class of $\mathbb{H}_{i}$ -representable matroids for $i\in\{1,2,3,4\}$ . We finish with some final remarks in Section 6, including the observation that each Hydra partial field is universal.

Our notation and terminology follows Oxley [18], unless specified otherwise. We note that some of the results of this paper have also appeared, informally, on the MatroidUnion blog²²2See http://matroidunion.org/?p=3815..

2. Preliminaries

For a matroid $M$ and a set of matroids $\mathcal{N}$ , we say that $M$ has an $\mathcal{N}$ -minor if $M$ has a minor isomorphic to $N$ for some $N\in\mathcal{N}$ .

Partial fields

A partial field is a pair $(R,G)$ , where $R$ is a commutative ring with unity, and $G$ is a subgroup of the group of units of $R$ such that $-1\in G$ . Note that $(\mathbb{F},\mathbb{F}^{*})$ is a partial field for any field $\mathbb{F}$ .

For disjoint sets $X$ and $Y$ , we refer to a matrix with rows labelled by elements of $X$ and columns labelled by elements of $Y$ as an $X\times Y$ matrix. Let $\mathbb{P}=(R,G)$ be a partial field, and let $A$ be an $X\times Y$ matrix with entries from $R$ . Then $A$ is a $\mathbb{P}$ -matrix if every square submatrix of $A$ with a non-zero determinant is in $G$ . If $X^{\prime}\subseteq X$ and $Y^{\prime}\subseteq Y$ , then we write $A[X^{\prime},Y^{\prime}]$ to denote the submatrix of $A$ with rows labelled by $X^{\prime}$ and columns labelled by $Y^{\prime}$ . We also say that $p$ is an element of $\mathbb{P}$ , and write $p\in\mathbb{P}$ , if $p\in G$ or $p=0$ .

Lemma 2.1 ([21, Theorem 2.8]).

Let $\mathbb{P}$ be a partial field, and let $A$ be an $X\times Y$ $\mathbb{P}$ -matrix, where $X$ and $Y$ are disjoint sets. Let

\mathcal{B}=\{X\}\cup\{X\triangle Z:|X\cap Z|=|Y\cap Z|,\det(A[X\cap Z,Y\cap Z])\neq 0\}.

Then $\mathcal{B}$ is the family of bases of a matroid on ground set $X\cup Y$ .

For an $X\times Y$ $\mathbb{P}$ -matrix $A$ , we let $M[A]$ denote the matroid in Lemma 2.1, and say that $A$ is a $\mathbb{P}$ -representation of $M[A]$ . Note that this is sometimes known as a reduced $\mathbb{P}$ -representation in the literature; here, all representations will be “reduced”, so we simply refer to them as representations. A matroid $M$ is $\mathbb{P}$ -representable if there exists some $\mathbb{P}$ -matrix $A$ such that $M\cong M[A]$ .

For partial fields $\mathbb{P}_{1}$ and $\mathbb{P}_{2}$ , a function $\phi:\mathbb{P}_{1}\rightarrow\mathbb{P}_{2}$ is a homomorphism if

(i)

$\phi(1)=1$ ,
(ii)

$\phi(pq)=\phi(p)\phi(q)$ for all $p,q\in\mathbb{P}_{1}$ , and
(iii)

$\phi(p)+\phi(q)=\phi(p+q)$ for all $p,q\in\mathbb{P}_{1}$ such that $p+q\in\mathbb{P}_{1}$ .

The existence of a homomorphism from $\mathbb{P}_{1}$ to $\mathbb{P}_{2}$ certifies that any $\mathbb{P}_{1}$ -representable matroid is also $\mathbb{P}_{2}$ -representable:

Lemma 2.2 ([21, Corollary 2.9]).

Let $\mathbb{P}_{1}$ and $\mathbb{P}_{2}$ be partial fields and let $\phi:\mathbb{P}_{1}\rightarrow\mathbb{P}_{2}$ be a homomorphism. If a matroid is $\mathbb{P}_{1}$ -representable, then it is also $\mathbb{P}_{2}$ -representable. In particular, if $[a_{ij}]$ is a $\mathbb{P}_{1}$ -representation of a matroid $M$ , then $[\phi(a_{ij})]$ is a $\mathbb{P}_{2}$ -representation of $M$ .

Representability over a partial field can be used to characterise representability over each field in a set of fields. Indeed, for any finite set of fields $\mathcal{F}$ , there exists a partial field $\mathbb{P}$ such that a matroid is $\mathcal{F}$ -representable if and only if it is $\mathbb{P}$ -representable [22, Corollary 2.20].

Let $M$ be a matroid. Pendavingh and van Zwam described [21, Section 4.2] a canonical construction of a partial field with the property that for every partial field $\mathbb{P}$ , the matroid $M$ is $\mathbb{P}$ -representable if and only if there exists a homomorphism $\phi:\mathbb{P}_{M}\rightarrow\mathbb{P}$ (see also [2]). We call the partial field $\mathbb{P}_{M}$ the universal partial field of $M$ .

Stabilizers

Let $\mathbb{P}=(R,G)$ be a partial field, and let $A$ and $A^{\prime}$ be $\mathbb{P}$ -matrices. We say that $A$ and $A^{\prime}$ are scaling equivalent if $A^{\prime}$ can be obtained from $A$ by scaling rows and columns by elements of $G$ . If $A^{\prime}$ can be obtained from $A$ by scaling, pivoting, and permuting rows and permuting columns (permuting labels at the same time), then we say that $A$ and $A^{\prime}$ are projectively equivalent. If $A^{\prime}$ can be obtained from $A$ by scaling, pivoting, permuting rows and permuting columns, and also applying automorphisms of $\mathbb{P}$ , then we say that $A$ and $A^{\prime}$ are algebraically equivalent.

We say that a matroid $M$ is uniquely $\mathbb{P}$ -representable if any two $\mathbb{P}$ -representations of $M$ are algebraically equivalent. On the other hand, two $\mathbb{P}$ -representations of $M$ are inequivalent if they are not algebraically equivalent.

Let $M$ and $N$ be $\mathbb{P}$ -representable matroids, where $M$ has an $N$ -minor. Then $N$ stabilizes $M$ over $\mathbb{P}$ if for any scaling-equivalent $\mathbb{P}$ -representations $A_{1}^{\prime}$ and $A_{2}^{\prime}$ of $N$ that extend to $\mathbb{P}$ -representations $A_{1}$ and $A_{2}$ of $M$ , respectively, $A_{1}$ and $A_{2}$ are scaling equivalent.

For a partial field $\mathbb{P}$ , let $\mathcal{M}(\mathbb{P})$ be the class of matroids representable over $\mathbb{P}$ . A matroid $N\in\mathcal{M}(\mathbb{P})$ is a $\mathbb{P}$ -stabilizer if, for any $3$ -connected matroid $M\in\mathcal{M}(\mathbb{P})$ having an $N$ -minor, the matroid $N$ stabilizes $M$ over $\mathbb{P}$ . Following Geelen et al. [13], we say that a matroid $N$ is a strong $\mathbb{P}$ -stabilizer if, for any $3$ -connected matroid $M\in\mathcal{M}(\mathbb{P})$ having an $N$ -minor, the matroid $N$ stabilizes $M$ over $\mathbb{P}$ , and every $\mathbb{P}$ -representation of $N$ extends to a $\mathbb{P}$ -representation of $M$ .

The Hydra- $i$ partial fields

The Hydra- $5$ partial field is:

\mathbb{H}_{5}=\left(\mathbb{Q}(\alpha,\beta,\gamma),\left<-1,\alpha,\beta,\gamma,\alpha-1,\beta-1,\gamma-1,\alpha-\gamma,\gamma-\alpha\beta,(1-\gamma)-(1-\alpha)\beta\right>\right)

where $\alpha,\beta,\gamma$ are indeterminates.

The Hydra- $4$ partial field is:

\mathbb{H}_{4}=\left(\mathbb{Q}(\alpha,\beta),\left<-1,\alpha,\beta,\alpha-1,\beta-1,\alpha\beta-1,\alpha+\beta-2\alpha\beta\right>\right)

where $\alpha,\beta$ are indeterminates.

The Hydra- $3$ partial field is:

\mathbb{H}_{3}=\left(\mathbb{Q}(\alpha),\left<-1,\alpha,1-\alpha,\alpha^{2}-\alpha+1\right>\right)

where $\alpha$ is an indeterminate.

The Hydra- $2$ partial field is:

\mathbb{H}_{2}=\left(\mathbb{Z}\left[i,1/2\right],\left<i,1-i\right>\right)

where $i$ is a root of $x^{2}+1=0$ .

We also let $\mathbb{H}_{1}=\mathrm{GF}(5)$ , and $\mathbb{H}_{6}=\mathbb{H}_{5}$ .

For each $i\in\{1,2,3,4\}$ , there is a homomorphism $\phi_{i}:\mathbb{H}_{i+1}\rightarrow\mathbb{H}_{i}$ . For example, let $\phi_{4}$ be determined by $\phi_{4}(\alpha)=\alpha$ , $\phi_{4}(\beta)=\frac{1}{1-\alpha}$ , and $\phi_{4}(\gamma)=\frac{\alpha(\beta-1)}{\beta(1-\alpha)}$ ; let $\phi_{3}$ be determined by $\phi_{3}(\alpha)=\alpha$ and $\phi_{4}(\beta)=\frac{\alpha}{\alpha-1}$ ; let $\phi_{2}$ be determined by $\phi_{2}(\alpha)=i$ ; and let $\phi_{1}$ be determined by $\phi_{1}(i)=2$ .

Lemma 2.3 ([21, Theorem 5.7]).

Let $M$ be a $\mathrm{GF}(5)$ -representable matroid that is $3$ -connected and has a $\{U_{2,5},U_{3,5}\}$ -minor, and let $i\in\{1,2,3,4,5,6\}$ . The matroid $M$ is $\mathbb{H}_{i}$ -representable if and only if $M$ has at least $i$ inequivalent representations over $\mathrm{GF}(5)$ .

For $\mathbb{H}_{2}$ -representable matroids that are $3$ -connected with a $\{U_{2,5},U_{3,5}\}$ -minor, the two inequivalent $\mathrm{GF}(5)$ -representations can be obtained by substituting $i\in\{2,3\}$ . For $\mathbb{H}_{3}$ -representable matroids, three inequivalent $\mathrm{GF}(5)$ -representations can be obtained by substituting $\alpha\in\{2,3,4\}$ . For $\mathbb{H}_{4}$ -representable matroids, four inequivalent $\mathrm{GF}(5)$ -representations can be obtained by substituting

(\alpha,\beta)\in\{(2,2),(3,3),(3,4),(4,3)\}.

For $\mathbb{H}_{5}$ -representable matroids, six inequivalent $\mathrm{GF}(5)$ -representations can be obtained by substituting

(\alpha,\beta,\gamma)\in\{(2,3,3),(3,2,2),(4,3,3),(2,4,4),(3,2,4),(4,4,2)\}.

Corollary 2.4 (see [21, Lemma 5.17]).

Let $M$ be a $\mathrm{GF}(5)$ -representable matroid that is $3$ -connected and has a $\{U_{2,5},U_{3,5}\}$ -minor. The matroid $M$ is $\mathbb{H}_{5}$ -representable if and only if $M$ has six inequivalent representations over $\mathrm{GF}(5)$ .

A matroid is dyadic if it is representable over $\mathrm{GF}(3)$ and $\mathrm{GF}(5)$ . Equivalently, a matroid is dyadic if it is $\mathbb{D}$ -representable, where $\mathbb{D}$ is the partial field

(\mathbb{Z}[1/2],\left<-1,2\right>).

There is a homomorphism from $\mathbb{D}$ to every field with characteristic not two. There is also a homomorphism from $\mathbb{D}$ to $\mathbb{H}_{2}$ . Note that, although a dyadic matroid is $\mathbb{H}_{2}$ -representable, it has no $\{U_{2,5},U_{3,5}\}$ -minor, so Lemma 2.3 does not apply.

The $k$ -regular partial fields

For a non-negative integer $k$ , the $k$ -regular partial field is:

\mathbb{U}_{k}=(\mathbb{Q}(\alpha_{1},\dots,\alpha_{k}),\left<\{x-y:x,y\in\{0,1,\alpha_{1},\dotsc,\alpha_{k}\}\textrm{ and }x\neq y\}\right>),

where $\alpha_{1},\dotsc,\alpha_{k}$ are indeterminates. A matroid is $k$ -regular if it is representable over the $k$ -regular partial field.

Note that $\mathbb{U}_{k}$ is the universal partial field of $U_{2,3+k}$ [26, Theorem 3.3.24]. It is easy to see that, for any non-negative integer $k$ , there is a homomorphism from $\mathbb{U}_{k}$ to $\mathbb{U}_{k+1}$ .

In particular, the $0$ -regular (or just regular) partial field is

\mathbb{U}_{0}=(\mathbb{Q},\{1,-1\});

the $1$ -regular (or near-regular) partial field is

\mathbb{U}_{1}=(\mathbb{Q}(\alpha),\left<-1,\alpha,\alpha-1\right>),

where $\alpha$ is an indeterminate; the $2$ -regular partial field is

\mathbb{U}_{2}=(\mathbb{Q}(\alpha_{1},\alpha_{2}),\left<-1,\alpha_{1},\alpha_{2},\alpha_{1}-1,\alpha_{2}-1,\alpha_{1}-\alpha_{2}\right>),

where $\alpha_{1},\alpha_{2}$ are indeterminates; and the $3$ -regular partial field is

\mathbb{U}_{3}=(\mathbb{Q}(\alpha_{1},\alpha_{2},\alpha_{3}),\left<-1,\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{1}-1,\alpha_{2}-1,\alpha_{3}-1,\alpha_{1}-\alpha_{2},\alpha_{1}-\alpha_{3},\alpha_{2}-\alpha_{3}\right>),

where $\alpha_{1},\alpha_{2},\alpha_{3}$ are indeterminates.

Van Zwam [26] observed that there is a homomorphism from $\mathbb{U}_{3}$ to $\mathbb{H}_{5}$ . It turns out that there is a homomorphism $\phi:\mathbb{U}_{3}\rightarrow\mathbb{H}_{5}$ where $\phi$ is a bijection [7].

Lemma 2.5 ([7, Lemma 2.25]).

The partial fields $\mathbb{H}_{5}$ and $\mathbb{U}_{3}$ are isomorphic. In particular, a matroid is $3$ -regular if and only if it is $\mathbb{H}_{5}$ -representable.

The next corollary then follows from Lemmas 2.3, 2.5 and 2.4.

Corollary 2.6.

Let $M$ be a $3$ -connected matroid with a $\{U_{2,5},U_{3,5}\}$ -minor. The following are equivalent:

(i)

$M$ is $3$ -regular.
(ii)

$M$ has at least five inequivalent $\mathrm{GF}(5)$ -representations.
(iii)

$M$ has precisely six inequivalent $\mathrm{GF}(5)$ -representations.

Alternatively, a matroid is $3$ -regular if and only if it is either near-regular or has six inequivalent $\mathrm{GF}(5)$ -representations.

Lemma 2.7 ([26, Lemmas 6.2.5 and 7.3.15]).

For $i\in\{1,2,3,4\}$ , let $N$ be an excluded minor for the class of $\mathbb{H}_{i+1}$ -representable matroids, such that $N$ is $\mathbb{H}_{i}$ -representable. Then $N$ is a strong $\mathbb{H}_{i}$ -stabilizer.

Other partial fields

Although not of central importance, we also refer to the following partial fields:

\mathbb{K}_{2}=(\mathbb{Q}(\alpha),\left<-1,\alpha-1,\alpha,\alpha+1\right>)

where $\alpha$ is an indeterminate, and

\mathbb{S}=(\mathbb{Z}[\zeta],\left<\zeta\right>),

where $\zeta$ is a root of $x^{2}-x+1=0$ . See [26] for further details.

Delta-wye exchange

Let $M$ be a matroid with a triangle $T=\{a,b,c\}$ . Consider a copy of $M(K_{4})$ having $T$ as a triangle with $\{a^{\prime},b^{\prime},c^{\prime}\}$ as the complementary triad labelled such that $\{a,b^{\prime},c^{\prime}\}$ , $\{a^{\prime},b,c^{\prime}\}$ and $\{a^{\prime},b^{\prime},c\}$ are triangles. Let $P_{T}(M,M(K_{4}))$ denote the generalised parallel connection of $M$ with this copy of $M(K_{4})$ along the triangle $T$ . Let $M^{\prime}$ be the matroid $P_{T}(M,M(K_{4}))\backslash T$ where the elements $a^{\prime}$ , $b^{\prime}$ and $c^{\prime}$ are relabelled as $a$ , $b$ and $c$ respectively. The matroid $M^{\prime}$ is said to be obtained from $M$ by a $\Delta$ - $Y$ exchange on the triangle $T$ , and is denoted $\Delta_{T}(M)$ . Dually, $M^{\prime\prime}$ is obtained from $M$ by a $Y$ - $\Delta$ exchange on the triad $T^{*}=\{a,b,c\}$ if $(M^{\prime\prime})^{*}$ is obtained from $M^{*}$ by a $\Delta$ - $Y$ exchange on $T^{*}$ . The matroid $M^{\prime\prime}$ is denoted $\nabla_{T^{*}}(M)$ .

We say that a matroid $M_{1}$ is $\Delta Y$ -equivalent to a matroid $M_{0}$ if $M_{1}$ can be obtained from $M_{0}$ by a sequence of $\Delta$ - $Y$ and $Y$ - $\Delta$ exchanges on coindependent triangles and independent triads, respectively. We let $\Delta(M)$ denote the set of matroids that are $\Delta Y$ -equivalent to $M$ . For a set of matroids $\mathcal{M}$ , we use $\Delta(\mathcal{M})$ to denote $\bigcup_{M\in\mathcal{M}}\Delta(M)$ . We also use $\Delta^{*}(M)$ to denote $\Delta(\{M,M^{*}\})$ , and use $\Delta^{*}(\mathcal{M})$ to denote $\bigcup_{M\in\mathcal{M}}\Delta^{*}(M)$ .

Oxley, Semple, and Vertigan proved that the set of excluded minors for $\mathbb{P}$ -representability is closed under $\Delta$ - $Y$ exchange (see also [1]), and, more generally, segment-cosegment exchange.

Proposition 2.8 ([19, Theorem 1.1]).

Let $\mathbb{P}$ be a partial field, and let $M$ be an excluded minor for the class of $\mathbb{P}$ -representable matroids. If $M^{\prime}\in\Delta^{*}(M)$ , then $M^{\prime}$ is an excluded minor for the class of $\mathbb{P}$ -representable matroids.

Proxies

Following [8], in order to simulate a matroid representation over a partial field, we use a representation over a finite field where we have constraints on the subdeterminants appearing in the representation. We briefly recap this theory (for a more detailed account, see [8, Section 3]).

Let $\mathbb{P}=(R,G)$ be a partial field. We say that $p\in\mathbb{P}$ is fundamental if $1-p\in\mathbb{P}$ . We denote the set of fundamentals of $\mathbb{P}$ by $\mathfrak{F}(\mathbb{P})$ . Let $A$ be a $\mathbb{P}$ -matrix. For a $\mathbb{P}$ -matrix $A^{\prime}$ , we write $A^{\prime}\preceq A$ if $A^{\prime}$ is a submatrix of a $\mathbb{P}$ -matrix that is projectively equivalent to $A$ . The cross ratios of $A$ are

\operatorname{Cr}(A)=\left\{p:\begin{bmatrix}1&1\\ p&1\end{bmatrix}\preceq A\right\}.

Let $F\subseteq\mathfrak{F}(\mathbb{P})$ . We say that the $\mathbb{P}$ -matrix $A$ is $F$ -confined if $\operatorname{Cr}(A)\subseteq F\cup\{0,1\}$ . If $A$ is an $F$ -confined $\mathbb{P}$ -matrix and $\phi:\mathbb{P}\rightarrow\mathbb{P}^{\prime}$ is a homomorphism, then $M[A]=M[\phi(A)]$ and

\operatorname{Cr}(\phi(A))\subseteq\phi(F),

so that $\phi(A)$ is an $\phi(F)$ -confined representation over $\mathbb{P}^{\prime}$ .

For a finite field $\mathbb{F}$ and homomorphism $\phi:\mathbb{P}\rightarrow\mathbb{F}$ , let $F_{\phi}=\phi(\mathfrak{F}(\mathbb{P}))$ . We say that $(\mathbb{F},\phi)$ is a proxy for $\mathbb{P}$ if for any $F_{\phi}$ -confined $\mathbb{F}$ -matrix $A$ , there exists a $\mathbb{P}$ -matrix $A^{\prime}$ such that $\phi(A^{\prime})$ is scaling-equivalent to $A$ . The following is a consequence of [8, Theorem 3.5].

Lemma 2.9.

Let $i\in\{2,3,4,5\}$ . There exists a prime $p$ and a homomorphism $\phi:\mathbb{H}_{i}\rightarrow\mathrm{GF}(p)$ such that $(\mathrm{GF}(p),\phi)$ is a proxy for $\mathbb{H}_{i}$ .

In particular, when $(\mathrm{GF}(p),\phi)$ is a proxy for $\mathbb{H}_{i}$ , a matroid $M$ is $\mathbb{H}_{i}$ -representable if and only if $M$ has an $F_{\phi}$ -confined $\mathrm{GF}(p)$ -representation. In Table 1, we provide the proxies for each Hydra- $i$ partial field that we used in our computations.

$\begin{array}[]{lll}\text{Partial field }&\text{Finite Field }&\text{Homomorphism}\\ \hline\cr\mathbb{H}_{2}&\mathrm{GF}(29)&i\mapsto 12\\ \mathbb{H}_{3}&\mathrm{GF}(151)&\alpha\mapsto 4\\ \mathbb{H}_{4}&\mathrm{GF}(947)&\alpha\mapsto 272,\beta\mapsto 928\\ \mathbb{H}_{5}&\mathrm{GF}(3527)&\alpha\mapsto 1249,\beta\mapsto 295,\gamma\mapsto 3517\\ \end{array}$

Table 1. Proxies for the Hydra-

i

partial fields.

Splitter theorems

For our computations, we keep track only of $3$ -connected matroids with an $\mathcal{N}$ -minor (where the matroids in $\mathcal{N}$ are strong stabilizers). Thus, we require the following consequence of Seymour’s Splitter Theorem (see, for example, [18, Corollary 12.2.2]).

Lemma 2.10 ([8, Corollary 2.8]).

Let $M$ be a $3$ -connected matroid with a proper $N$ -minor, where $N$ is not near-regular. Then, for some $\{M^{\prime},N^{\prime}\}\in\{(M,N),(M^{*},N^{*})\}$ , there exists an element $e\in E(M^{\prime})$ such that $M^{\prime}\backslash e$ is $3$ -connected and has an $N^{\prime}$ -minor.

Hall, Mayhew, and van Zwam [15] proved an excluded-minor characterisation for near-regular matroids. Two of the excluded minors for this class are $U_{2,5}$ and $U_{3,5}$ . In this paper, we deal primarily with $3$ -connected matroids that have a $\{U_{2,5},U_{3,5}\}$ -minor, so Lemma 2.10 applies.

To find the excluded minors for $\mathbb{H}_{5}$ -representability on up to $13$ elements, we use the technique referred to as “splicing” in [8, Section 4.5]. Briefly, when $M^{\prime}$ is a matroid, $M_{e}$ is an extension of $M^{\prime}$ by an element $e$ , and $M_{f}$ is an extension of $M^{\prime}$ by an element $f$ , then the extensions $M_{e}$ and $M_{f}$ are compatible if there exists a matroid $M$ on $E(M^{\prime})\cup\{e,f\}$ such that $M\backslash f=M_{e}$ and $M\backslash e=M_{f}$ . (A characterisation of compatible extensions was given by Cheung [11].) In this case, following [8], we say that $M$ is a splice of $M_{e}$ and $M_{f}$ . (We note that a “splice” has a different meaning in [4].) For a positive integer $n$ , an $(n+1)$ -element matroid can be constructed as the splice of two $n$ -element matroids, $M_{e}$ and $M_{f}$ say, where $M_{e}$ and $M_{f}$ are both single-element extensions of some $(n-1)$ -element matroid $M^{\prime}$ .

Starting from some set $\mathcal{N}$ of “seed matroids”, we build a catalog of the $3$ -connected matroids in the class that have an $N$ -minor for some $N\in\mathcal{N}$ . Suppose $M$ is a matroid that should appear in the catalog, where $M$ has an $N$ -minor for $N\in\mathcal{N}$ . In order to ensure that $M$ is obtained as the splice of two matroids $M_{a}$ and $M_{b}$ , we require a guarantee that, for some pair $\{a,b\}\subseteq E(M)$ , the matroid $M\backslash a,b$ is also in the catalog. In fact, by closing the catalog under duality, it suffices that one of $M$ and $M^{*}$ appears in the catalog. Thus, we require a pair $\{a,b\}\subseteq E(M)$ , such that either $M\backslash a,b$ or $M/a,b$ is $3$ -connected and has an $N$ -minor; such a pair $\{a,b\}$ is known as an $N$ -detachable pair [9].

To describe the conditions under which $M$ has no $N$ -detachable pairs, we require some definitions.

Definition 2.11.

Let $P$ be an exactly $3$ -separating set in a matroid $M$ , with $|P|=2t\geq 6$ . Suppose $P$ has the following properties:

(a)

there is a partition $\{L_{1},\dotsc,L_{t}\}$ of $P$ into pairs such that for all distinct $i,j\in\{1,\dotsc,t\}$ , the set $L_{i}\cup L_{j}$ is a cocircuit,
(b)

there is a partition $\{K_{1},\dotsc,K_{t}\}$ of $P$ into pairs such that for all distinct $i,j\in\{1,\dotsc,t\}$ , the set $K_{i}\cup K_{j}$ is a circuit,
(c)

$M/p$ and $M\backslash p$ are $3$ -connected for each $p\in P$ ,
(d)

for all distinct $i,j\in\{1,\dotsc,t\}$ , the matroid $\operatorname{si}(M/a/b)$ is $3$ -connected for any $a\in L_{i}$ and $b\in L_{j}$ , and
(e)

for all distinct $i,j\in\{1,\dotsc,t\}$ , the matroid $\operatorname{co}(M\backslash a\backslash b)$ is $3$ -connected for any $a\in K_{i}$ and $b\in K_{j}$ .

Then we say $P$ is a spiky $3$ -separator of $M$ .

Definition 2.12.

A matroid $M$ is a quad-flower if $r(M)=r^{*}(M)=6$ , and there is a labelling $\bigcup_{\ell\in\{1,2,3\}}\{p_{\ell},q_{\ell},s_{\ell},t_{\ell}\}$ of $E(M)$ such that, for all $(i,j)\in\{(1,2),(2,3),(3,1)\}$ ,

(i)

$\{p_{i},q_{i},s_{i},t_{i}\}$ is a circuit and a cocircuit,
(ii)

the sets $\{p_{i},q_{i},p_{j},s_{j}\}$ , $\{p_{i},q_{i},q_{j},t_{j}\}$ , $\{s_{i},t_{i},p_{j},s_{j}\}$ , and $\{s_{i},t_{i},q_{j},t_{j}\}$ are circuits, and
(iii)

the sets $\{p_{i},s_{i},p_{j},q_{j}\}$ , $\{p_{i},s_{i},s_{j},t_{j}\}$ , $\{q_{i},t_{i},p_{j},q_{j}\}$ , and $\{q_{i},t_{i},s_{j},t_{j}\}$ are cocircuits.

We call $\bigcup_{\ell\in\{1,2,3\}}\{p_{\ell},q_{\ell},s_{\ell},t_{\ell}\}$ the quad-flower labelling of $M$ , and say that $M$ has standard labelling if $E(M)=\bigcup_{\ell\in\{1,2,3\}}\{p_{\ell},q_{\ell},s_{\ell},t_{\ell}\}$ and the quad-flower labelling is obtained by applying the identity map.

An example of a quad-flower is given in Figure 1.

Figure 1. A quad-flower.

Definition 2.13.

A matroid $M$ is a nest if $r(M)=r^{*}(M)=6$ , and there is a labelling $\{e_{1},e_{1}^{\prime},e_{2},e_{2}^{\prime},\dotsc,e_{6},e_{6}^{\prime}\}$ of $E(M)$ such that

(i)

the following sets are circuits:

\{e_{1},e_{2},e_{3}^{\prime},e_{4}^{\prime}\},\{e_{3},e_{4},e_{5}^{\prime},e_{6}^{\prime}\},\{e_{5},e_{6},e_{1}^{\prime},e_{2}^{\prime}\},

\{e_{5},e_{4},e_{1}^{\prime},e_{3}^{\prime}\},\{e_{1},e_{3},e_{2}^{\prime},e_{6}^{\prime}\},\{e_{2},e_{6},e_{5}^{\prime},e_{4}^{\prime}\},

\{e_{2},e_{3},e_{5}^{\prime},e_{1}^{\prime}\},\{e_{5},e_{1},e_{4}^{\prime},e_{6}^{\prime}\},\{e_{4},e_{6},e_{2}^{\prime},e_{3}^{\prime}\},

\{e_{4},e_{1},e_{2}^{\prime},e_{5}^{\prime}\},\{e_{2},e_{5},e_{3}^{\prime},e_{6}^{\prime}\},\{e_{3},e_{6},e_{4}^{\prime},e_{1}^{\prime}\},

\{e_{3},e_{5},e_{4}^{\prime},e_{2}^{\prime}\},\{e_{4},e_{2},e_{1}^{\prime},e_{6}^{\prime}\},\{e_{1},e_{6},e_{3}^{\prime},e_{5}^{\prime}\},

(ii)

and the following sets are cocircuits:

\{e_{3},e_{4},e_{1}^{\prime},e_{2}^{\prime}\},\{e_{1},e_{2},e_{5}^{\prime},e_{6}^{\prime}\},\{e_{5},e_{6},e_{3}^{\prime},e_{4}^{\prime}\}

\{e_{1},e_{3},e_{5}^{\prime},e_{4}^{\prime}\},\{e_{5},e_{4},e_{2}^{\prime},e_{6}^{\prime}\},\{e_{2},e_{6},e_{1}^{\prime},e_{3}^{\prime}\}

\{e_{5},e_{1},e_{2}^{\prime},e_{3}^{\prime}\},\{e_{2},e_{3},e_{4}^{\prime},e_{6}^{\prime}\},\{e_{4},e_{6},e_{5}^{\prime},e_{1}^{\prime}\}

\{e_{2},e_{5},e_{4}^{\prime},e_{1}^{\prime}\},\{e_{4},e_{1},e_{3}^{\prime},e_{6}^{\prime}\},\{e_{3},e_{6},e_{2}^{\prime},e_{5}^{\prime}\}

\{e_{4},e_{2},e_{3}^{\prime},e_{5}^{\prime}\},\{e_{3},e_{5},e_{1}^{\prime},e_{6}^{\prime}\},\{e_{1},e_{6},e_{4}^{\prime},e_{2}^{\prime}\}.

We call $\{e_{1},e_{1}^{\prime},e_{2},e_{2}^{\prime},\dotsc,e_{6},e_{6}^{\prime}\}$ the nest labelling of $M$ , and say that $M$ has standard labelling if $E(M)=\{e_{1},e_{1}^{\prime},e_{2},e_{2}^{\prime},\dotsc,e_{6},e_{6}^{\prime}\}$ and the nest labelling is obtained by applying the identity map.

Note that a “nest” is referred to as a “nest of twisted cubes” in [9].

Theorem 2.14 ([9, Corollary 6.3]).

Let $M$ be a $3$ -connected matroid and let $N$ be a $3$ -connected minor of $M$ with $|E(M)|\geq 11$ , $|E(N)|\geq 4$ , and $|E(M)|-|E(N)|\geq 5$ . Then either

(i)

$M$ has an $N$ -detachable pair;
(ii)

there is a matroid $M^{\prime}$ obtained by performing a single $\Delta$ - $Y$ or $Y$ - $\Delta$ exchange on $M$ such that $M^{\prime}$ has an $N$ -minor and an $N$ -detachable pair, or
(iii)

$M$ has a spiky $3$ -separator $P$ such that at most one element of $E(M)-E(N)$ is not in $P$ , or
(iv)

$|E(M)|=12$ , and $M$ is either a quad-flower or a nest.

The next lemma is used to handle the possibility that Theorem 2.14(iii) holds. It is a slight generalisation of [6, Lemma 7.2], but can be proved using essentially the same argument as given there.

Lemma 2.15 (see [6, Lemma 7.2]).

Let $\mathbb{P}$ be a partial field, let $N$ be a non-binary $3$ -connected strong stabilizer for the class of $\mathbb{P}$ -representable matroids, and let $M$ be an excluded minor for the class of $\mathbb{P}$ -representable matroids, where $M$ has an $N$ -minor. If $M$ has a spiky $3$ -separator $P$ such that at most one element of $E(M)-E(N)$ is not in $P$ , then $|E(M)|\leq|E(N)|+5$ .

To handle the possibility that Theorem 2.14(iv) holds, we need to consider what (partial) fields a quad-flower, or a nest, is representable over. This analysis is done in the next section.

3. Exceptional $12$ -element matroids

When $M$ is a quad-flower or a nest, and $N$ is a $3$ -connected minor of $M$ , we cannot guarantee that $M$ has an $N$ -detachable pair. When enumerating potential excluded minors using splicing, these matroids will be skipped. Thus, we need to be able to verify that these matroids are not excluded minors for the class of $\mathbb{H}_{5}$ -representable matroids. It suffices to verify that each of these matroids contains either the Fano matroid $F_{7}$ , the non-Fano matroid $F_{7}^{-}$ , or the relaxation of the non-Fano matroid $F_{7}^{=}$ , as a proper minor (due to the upcoming Lemma 4.2). However, so that these techniques can also be applied more generally (for partial fields other than $\mathbb{H}_{5}$ ), we briefly study quad-flowers and nests. We will show that there are, up to isomorphism, three quad-flowers that are representable over some field (one is binary, one is dyadic, and one is representable over all fields of size at least four), whereas any other quad-flower has a proper minor that is not representable over any field. Up to isomorphism, there are three nests: one is binary, one is dyadic, while the other is not representable over any field.

Quad-flowers

First, we consider the circuits in a quad-flower. We will see that the only sets that can be non-spanning circuits are the $4$ -element circuits listed in the definition, and certain $6$ -element sets. Among these $6$ -element sets, there are some that are circuits in every quad-flower, and some that can be either a circuit or a basis. We introduce notation to refer to the latter sets. Recall that a quad-flower with standard labelling is on the ground set $\bigcup_{\ell\in\{1,2,3\}}\{p_{\ell},q_{\ell},s_{\ell},t_{\ell}\}$ . For $J\subseteq\{1,2,3\}$ , let $X_{J}$ be the $6$ -element set such that $\{p_{i},t_{i}\}\subseteq X_{J}$ for each $i\in J$ , and $\{q_{i},s_{i}\}\subseteq X_{J}$ for each $i\in\{1,2,3\}-J$ .

Lemma 3.1.

Let $M$ be a quad-flower with standard labelling. Then all of the following hold:

(i)

If $C$ is a non-spanning circuit of $M$ , then $|C|\in\{4,6\}$ .
(ii)

If $C$ is a circuit of $M$ with $|C|=4$ , then $C$ is one of the $15$ circuits listed in Definition 2.12(i)–(ii).
(iii)

For any $(i,j,k)\in\{(1,2,3),(2,3,1),(3,1,2)\}$ and $Z_{i}\in\{\{p_{i},q_{i}\},\{s_{i},t_{i}\}\}$ , $Z_{j}\in\{\{p_{j},t_{j}\},\{q_{j},s_{j}\}\}$ , and $Z_{k}\in\{\{p_{k},s_{k}\},\{q_{k},t_{k}\}\}$ , the set $Z_{i}\cup Z_{j}\cup Z_{k}$ is a $6$ -element circuit of $M$ .
(iv)

If $C$ is a circuit of $M$ with $|C|=6$ but $C$ is not a circuit described in (iii), then $C=X_{J}$ for some $J\subseteq\{1,2,3\}$ .

Proof.

By orthogonality and the fact that no circuit is properly contained in another, a quad-flower has no circuits of size at most five other than the $15$ circuits of size four listed in the definition. Since $r(M)=6$ , this proves (i) and (ii). Moreover, by circuit elimination and orthogonality, it follows that (iii) holds. For (iv), let $X\subseteq E(M)$ with $|X|=6$ such that $X$ does not contain one of the $15$ circuits of size four, and $X$ is not a circuit as described in (iii). It follows that $X=X_{J}$ for some $J\subseteq\{1,2,3\}$ . ∎

Although we focus on the circuits in a quad-flower, a similar statement to Lemma 3.1 can be made regarding the cocircuits. It then follows easily that a quad-flower is $3$ -connected.

We distinguish three particular quad-flowers; we will show that (up to isomorphism) these are the only three quad-flowers that are representable over some field. We say that

•

$M$ is a binary quad-flower if $M$ has a quad-flower labelling such that $X_{J}$ is a circuit for all $J\subseteq\{1,2,3\}$ ;
•

$M$ is a dyadic quad-flower if $M$ has a quad-flower labelling such that, for $J\subseteq\{1,2,3\}$ , the set $X_{J}$ is a circuit if and only if $|J|\in\{0,2\}$ ; and
•

$M$ is a free quad-flower if $M$ has a quad-flower labelling such that $X_{J}$ is a basis for all $J\subseteq\{1,2,3\}$ .

The next lemma illustrates the reasoning behind the first two of these names.

Lemma 3.2.

Let $M$ be a quad-flower. Then precisely one of the following holds:

(i)

$M$ is a binary quad-flower, and $M$ is $\mathbb{F}$ -representable if and only if $\mathbb{F}$ has characteristic two,
(ii)

$M$ is a dyadic quad-flower, and $M$ is $\mathbb{F}$ -representable if and only if $\mathbb{F}$ has characteristic not two,
(iii)

$M$ is a free quad-flower, and $M$ is $\mathbb{F}$ -representable if and only if $|\mathbb{F}|\geq 4$ , or
(iv)

$M$ has a proper minor $M^{\prime}$ such that $M^{\prime}$ is not representable over any field.

Proof.

We let $M$ be a quad-flower with standard labelling, so Lemma 3.1 holds. Consider $X_{J}$ for some $J\subseteq\{1,2,3\}$ . There are $8$ possibilities for the set $X_{J}$ , and each of these could be either a circuit or a basis in a quad-flower $M$ . It is convenient to think of each $X_{J}$ as a vertex in a graph: consider the $3$ -dimensional hypercube graph $Q$ on vertex set $\{X_{J}:J\subseteq\{1,2,3\}\}$ , where distinct vertices $X_{I}$ and $X_{J}$ are adjacent if and only if the cardinality of the set difference $I\triangle J$ is one. Let $\mathbf{X}$ be a subset of the vertices of $Q$ , where $X_{J}\in\mathbf{X}$ if $X_{J}$ is a circuit of $M$ , whereas $X_{J}\notin\mathbf{X}$ if $X_{J}$ is a basis of $M$ . Then, for the quad-flower $M$ , we obtain an auxiliary set $\mathbf{X}\subseteq V(Q)$ .

Suppose that $X_{I},X_{J}\in\mathbf{X}$ for some $X_{I}$ and $X_{J}$ that are adjacent in $Q$ (so $|I\triangle J|=1$ ). Let $Z\subseteq X_{I}\cap X_{J}$ such that $|Z|=3$ . Then it follows that $M/Z$ has an $F_{7}$ -restriction (where $F_{7}$ is the Fano matroid), so $M$ is not representable over any field that does not have characteristic two. On the other hand, if precisely one of $X_{I}$ and $X_{J}$ is in $\mathbf{X}$ , then it follows that $M/Z$ has an $F^{-}_{7}$ -restriction (where $F^{-}_{7}$ is the non-Fano matroid), so $M$ is not representable over fields with characteristic two.

We first assume that $M$ is not a binary quad-flower, a dyadic quad-flower, nor a free quad-flower, and show that (iv) holds. In particular, since $M$ is not a binary quad-flower nor a free quad-flower, $\mathbf{X}\notin\{\emptyset,V(Q)\}$ . We claim that if $\mathbf{X}$ is not a stable set in $Q$ , then $M$ has a proper minor that is not representable over any field. Since $\mathbf{X}\notin\{\emptyset,V(Q)\}$ , there exist adjacent vertices of $Q$ , precisely one of which is in $\mathbf{X}$ . Suppose $\mathbf{X}$ is not a stable set in $Q$ , so there exist adjacent vertices of $Q$ in $\mathbf{X}$ . Then there exist distinct $I,J,K\subseteq\{1,2,3\}$ such that $X_{I},X_{J}\in\mathbf{X}$ and $X_{K}\notin\mathbf{X}$ , where $X_{J}$ is adjacent to both $X_{I}$ and $X_{K}$ in $Q$ . It now follows, by the previous paragraph, that $M$ has a proper minor that is not representable over any field, as claimed.

Now we assume that $\mathbf{X}$ is a stable set in $Q$ . We note that $M$ is one of four quad-flowers, up to isomorphism (recalling that $M$ is not a dyadic or free quad-flower). For any $j\in\{1,2,3\}$ , the set $\mathbf{Y}_{j}=\{X_{J}:j\in J\}$ or the set $\mathbf{Y}^{\prime}_{j}=\{X_{J}:j\notin J\}$ induces a $4$ -cycle in $Q$ , so at most two of the four sets in $\mathbf{Y}_{j}$ or $\mathbf{Y}^{\prime}_{j}$ are circuits. Since $M$ is not a binary quad-flower, a dyadic quad-flower, nor a free quad-flower, there exists some $j\in\{1,2,3\}$ such that either $|\mathbf{Y}_{j}\cap\mathbf{X}|=1$ or $|\mathbf{Y}^{\prime}_{j}\cap\mathbf{X}|=1$ . Up to isomorphism, we may assume that $\{p_{1},t_{1},p_{2},t_{2},p_{3},t_{3}\}$ is a circuit in $M$ , but the sets $\{p_{1},t_{1},p_{2},t_{2},q_{3},s_{3}\}$ , $\{p_{1},t_{1},q_{2},s_{2},p_{3},t_{3}\}$ , and $\{p_{1},t_{1},q_{2},s_{2},q_{3},s_{3}\}$ are bases. Consider $M^{\prime}=M/p_{1}$ . We claim that $M^{\prime}$ is not representable over any field. If $M^{\prime}$ has an $\mathbb{F}$ -representation $A^{\prime}$ for some field $\mathbb{F}$ , then it is easy to verify (see [18, Theorem 6.4.7], for example) that $A^{\prime}$ is projectively equivalent to a matrix of the form

\hbox{}\vbox{\kern 0.86108pt\hbox{$\kern 0.0pt\kern 2.5pt\kern-5.0pt\left[\kern 0.0pt\kern-2.5pt\kern-5.55557pt\vbox{\kern-0.86108pt\vbox{\vbox{ \halign{\kern\arraycolsep\hfil\@arstrut$\kbcolstyle#$\hfil\kern\arraycolsep& \kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep&& \kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep\cr 5.0pt\hfil\@arstrut$\scriptstyle$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle s_{1}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle t_{1}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle s_{2}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle t_{2}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle s_{3}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle t_{3}\\q_{1}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle-1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0\\p_{2}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle*$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1\\q_{2}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle*$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle*\\p_{3}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0\\q_{3}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle*$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle*$\hfil\kern 5.0pt\crcr}}}}\right]$}},

where $*$ signifies the entry is non-zero.

By considering the dependent sets $\{s_{1},s_{3},t_{3},q_{1},q_{3}\}$ , $\{s_{1},t_{1},s_{3},q_{1},p_{2}\}$ , $\{t_{1},s_{3},t_{3},q_{1},p_{2}\}$ , and $\{s_{2},t_{2},s_{3},p_{3},q_{3}\}$ , we see that the matrix is in fact of the form

\hbox{}\vbox{\kern 0.86108pt\hbox{$\kern 0.0pt\kern 2.5pt\kern-5.0pt\left[\kern 0.0pt\kern-2.5pt\kern-5.55557pt\vbox{\kern-0.86108pt\vbox{\vbox{ \halign{\kern\arraycolsep\hfil\@arstrut$\kbcolstyle#$\hfil\kern\arraycolsep& \kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep&& \kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep\cr 5.0pt\hfil\@arstrut$\scriptstyle$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle s_{1}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle t_{1}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle s_{2}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle t_{2}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle s_{3}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle t_{3}\\q_{1}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle-1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0\\p_{2}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\alpha$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1\\q_{2}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle-\alpha$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1\\p_{3}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0\\q_{3}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle-1$\hfil\kern 5.0pt\crcr}}}}\right]$}},

for some $\alpha\in\mathbb{F}-\{0\}$ . However, $\alpha=1$ since $\{t_{1},p_{2},t_{2},p_{3},t_{3}\}$ is a circuit in $M^{\prime}$ , but $\alpha\neq 1$ since $\{t_{1},q_{2},s_{2},q_{3},s_{3}\}$ is a basis in $M^{\prime}$ . So $M^{\prime}$ is not representable over any field.

We have shown that if $M$ is not a binary quad-flower, a dyadic quad-flower, nor a free quad-flower, then (iv) holds. Suppose now that $M$ is a quad-flower that is representable over some partial field $\mathbb{P}$ . Let $A$ be a $\mathbb{P}$ -representation for $M$ . Then it is easy to verify that $A$ is projectively equivalent to the matrix

\hbox{}\vbox{\kern 0.86108pt\hbox{$\kern 0.0pt\kern 2.5pt\kern-5.0pt\left[\kern 0.0pt\kern-2.5pt\kern-5.55557pt\vbox{\kern-0.86108pt\vbox{\vbox{ \halign{\kern\arraycolsep\hfil\@arstrut$\kbcolstyle#$\hfil\kern\arraycolsep& \kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep&& \kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep\cr 5.0pt\hfil\@arstrut$\scriptstyle$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle s_{1}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle t_{1}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle s_{2}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle t_{2}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle s_{3}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle t_{3}\\p_{1}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0\\q_{1}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle-1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0\\p_{2}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\alpha$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1\\q_{2}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle-\alpha$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1\\p_{3}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0\\q_{3}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle-1$\hfil\kern 5.0pt\crcr}}}}\right]$}},

where $\alpha\neq 0$ .

When $M$ is a binary quad-flower, we obtain a $\mathrm{GF}(2^{n})$ -representation (for any positive integer $n$ ) by setting $\alpha=1$ , so $M$ is representable over any field with characteristic two. Since a binary quad-flower has an $F_{7}$ -minor, it is not representable over any other field, so (i) holds.

When $M$ is a dyadic quad-flower, we obtain a $\mathbb{D}$ -representation by setting $\alpha=1$ (where $\det(A[\{p_{1},q_{2},q_{3}\},\{s_{1},t_{2},t_{3}\}])=2$ ). In particular, this is an $\mathbb{F}$ -representation for any field $\mathbb{F}$ having characteristic not two. Since a dyadic quad-flower has an $F_{7}^{-}$ -minor, (ii) holds.

When $M$ is a free quad-flower, then $M$ has an $F_{7}^{=}$ -minor, where $F_{7}^{=}$ is the matroid obtained from $F_{7}^{-}$ by relaxing a circuit-hyperplane. Thus $M$ is not $2$ -regular. In particular, as $F_{7}^{=}$ has a $U_{2,5}$ -minor, $M$ is not binary or ternary. However, $M$ is $\mathbb{F}$ -representable whenever $|\mathbb{F}|\geq 4$ , as then the above matrix is an $\mathbb{F}$ -representation after choosing $\alpha\in\mathbb{F}-\{0,1,-1\}$ . (Alternatively, the above matrix is a $\mathbb{K}_{2}$ -representation, and there is a homomorphism from $\mathbb{K}_{2}$ to any field with size at least four [26, Lemma 2.5.33].) This shows that (iii) holds. ∎

Nests

Next, we consider nests. We start by considering the circuits. We omit the proof of the next lemma; it follows from orthogonality, the circuit axioms, and the fact that $r(M)=6$ , similar to Lemma 3.1.

Lemma 3.3.

Let $M$ be a nest with standard labelling. Then all of the following hold:

(i)

If $C$ is a non-spanning circuit of $M$ , then $|C|\in\{4,6\}$ .
(ii)

If $C$ is a circuit of $M$ with $|C|=4$ , then $C$ is one of the $15$ circuits listed in Definition 2.13(i).
(iii)

If $\{e_{i},e_{j},e_{k}^{\prime},e_{\ell}^{\prime}\}$ is a circuit of $M$ , then $\{e_{i},e_{j}\}\cup(\{e_{1}^{\prime},\dotsc,e_{6}^{\prime}\}-\{e_{k}^{\prime},e_{\ell}^{\prime}\})$ and $(\{e_{1},\dotsc,e_{6}\}-\{e_{i},e_{j}\})\cup\{e_{k}^{\prime},e_{\ell}^{\prime}\}$ are $6$ -element circuits of $M$ .
(iv)

If $C$ is a circuit of $M$ with $|C|=6$ but $C$ is not a circuit described in (iii), then $C\in\{\{e_{1},e_{2},e_{3},e_{4},e_{5},e_{6}\},\{e_{1}^{\prime},e_{2}^{\prime},e_{3}^{\prime},e_{4}^{\prime},e_{5}^{\prime},e_{6}^{\prime}\}\}$ .

By Lemma 3.3, there are three non-isomorphic nests, depending on whether both, none, or one of the sets described in (iv) are circuits. We say that a nest is

•

tight if it has a nest labelling such that both $\{e_{1},e_{2},e_{3},e_{4},e_{5},e_{6}\}$ and $\{e_{1}^{\prime},e_{2}^{\prime},e_{3}^{\prime},e_{4}^{\prime},e_{5}^{\prime},e_{6}^{\prime}\}$ are circuits;
•

loose if it has a nest labelling such that both $\{e_{1},e_{2},e_{3},e_{4},e_{5},e_{6}\}$ and $\{e_{1}^{\prime},e_{2}^{\prime},e_{3}^{\prime},e_{4}^{\prime},e_{5}^{\prime},e_{6}^{\prime}\}$ are bases; and
•

slack if it has a nest labelling such that one of $\{e_{1},e_{2},e_{3},e_{4},e_{5},e_{6}\}$ and $\{e_{1}^{\prime},e_{2}^{\prime},e_{3}^{\prime},e_{4}^{\prime},e_{5}^{\prime},e_{6}^{\prime}\}$ is a circuit, and the other is a basis.

Lemma 3.4.

Let $M$ be a nest. Then $M$ has a $\{F_{7},F_{7}^{-}\}$ -minor. Moreover,

(i)

when $M$ is tight, $M$ is $\mathbb{F}$ -representable if and only if $\mathbb{F}$ has characteristic two;
(ii)

when $M$ is loose, $M$ is $\mathbb{F}$ -representable if and only if $\mathbb{F}$ has characteristic not two; and
(iii)

if $M$ is slack, then $M$ is not representable over any field.

Proof.

Assume that $M$ has standard labelling, and consider the minor $M_{1}=M/\{e_{4},e_{5},e_{6}\}\backslash\{e_{1}^{\prime},e_{2}^{\prime}\}$ . By Definition 2.13(i) and Lemma 3.3(iii), $M_{1}$ has circuits $\{e_{3},e_{3}^{\prime},e_{4}^{\prime}\}$ , $\{e_{3},e_{5}^{\prime},e_{6}^{\prime}\}$ , $\{e_{2},e_{5}^{\prime},e_{4}^{\prime}\}$ , $\{e_{1},e_{4}^{\prime},e_{6}^{\prime}\}$ , $\{e_{2},e_{3}^{\prime},e_{6}^{\prime}\}$ , and $\{e_{1},e_{3}^{\prime},e_{5}^{\prime}\}$ . Moreover, $\{e_{1},e_{2},e_{3}\}$ is either a circuit or a basis in $M_{1}$ . It follows that $M$ has an $\{F_{7},F_{7}^{-}\}$ -minor. Suppose that $M$ is tight or loose. If $M$ is tight, then $M$ has an $F_{7}$ -minor, so $M$ is not representable over any field that does not have characteristic two; and if $M$ is loose, then $M$ has an $F_{7}^{-}$ -minor, so $M$ is not representable over any field that has characteristic two. Let $A$ be the matrix

\kern 0.0pt\kern 2.5pt\kern-5.0pt\left[\kern 0.0pt\kern-2.5pt\kern-5.55557pt\vbox{\kern-0.86108pt\vbox{\vbox{ \halign{\kern\arraycolsep\hfil\@arstrut$\kbcolstyle#$\hfil\kern\arraycolsep& \kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep&& \kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep\cr 5.0pt\hfil\@arstrut$\scriptstyle$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle e_{1}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle e_{3}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle e_{5}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle e_{2}^{\prime}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle e_{4}^{\prime}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle e_{6}^{\prime}\\e_{2}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1\\e_{4}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1\\e_{6}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0\\e_{1}^{\prime}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1\\e_{3}^{\prime}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0\\e_{5}^{\prime}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt\crcr}}}}\right]

over a field $\mathbb{F}$ . When $\mathbb{F}$ has characteristic two, $A$ is an $\mathbb{F}$ -representation of a tight nest; otherwise, $A$ is an $\mathbb{F}$ -representation of a loose nest. This proves (i) and (ii).

It remains to prove (iii). Suppose $M$ is slack, and consider the minor $M_{2}=M/\{e_{1}^{\prime},e_{2}^{\prime},e_{3}^{\prime}\}\backslash\{e_{4},e_{6}\}$ . By Definition 2.13(i) and Lemma 3.3(iii), $M_{2}$ has circuits $\{e_{5},e_{1},e_{5}^{\prime}\}$ , $\{e_{1},e_{2},e_{4}^{\prime}\}$ , $\{e_{1},e_{3},e_{6}^{\prime}\}$ , $\{e_{2},e_{3},e_{5}^{\prime}\}$ , $\{e_{2},e_{5},e_{6}^{\prime}\}$ , and $\{e_{3},e_{5},e_{4}^{\prime}\}$ . Either $\{e_{1},e_{2},e_{3}\}$ is a circuit in $M_{1}$ and $\{e_{4}^{\prime},e_{5}^{\prime},e_{6}^{\prime}\}$ is a basis in $M_{2}$ , or $\{e_{1},e_{2},e_{3}\}$ is a basis in $M_{1}$ and $\{e_{4}^{\prime},e_{5}^{\prime},e_{6}^{\prime}\}$ is a circuit in $M_{2}$ . It follows that $M$ has both an $F_{7}$ - and $F_{7}^{-}$ -minor, so $M$ is not representable over any field. ∎

4. Excluded minors for $3$ -regular matroids

In this section, we consider excluded minors for the class of $\mathbb{H}_{5}$ -representable matroids. Recall that this is, equivalently, the class of $3$ -regular matroids. The following theorem is the recently obtained excluded-minor characterisation of $2$ -regular matroids [7, 8]. The matroids appearing in this theorem are well known (as they appear in the excluded-minor characterisation for either $\mathrm{GF}(4)$ -representable matroids [14] or near-regular matroids [15]), with two exceptions. The matroid $P_{8}^{-}$ is obtained from $P_{8}$ by relaxing one of the two disjoint circuit-hyperplanes. The matroid $TQ_{8}$ is a rank- $4$ sparse paving matroid on ground set $\{0,1,\dotsc,7\}$ with eight non-spanning circuits $\big{\{}\{i,i+2,i+4,i+5\}:i\in\{0,1,\dotsc,7\}\big{\}}$ , working modulo 8.

Theorem 4.1 ([7, Theorem 1.2]).

A matroid $M$ is $2$ -regular if and only if $M$ has no minor isomorphic to $U_{2,6}$ , $U_{3,6}$ , $U_{4,6}$ , $P_{6}$ , $F_{7}$ , $F_{7}^{*}$ , $F_{7}^{-}$ , $(F_{7}^{-})^{*}$ , $F_{7}^{=}$ , $(F_{7}^{=})^{*}$ , $\mathit{AG}(2,3)\backslash e$ , $(\mathit{AG}(2,3)\backslash e)^{*}$ , $(\mathit{AG}(2,3)\backslash e)^{\Delta Y}$ , $P_{8}$ , $P_{8}^{-}$ , $P_{8}^{=}$ , and $\mathit{TQ}_{8}$ .

The next lemma follows from Theorem 4.1, by considering which of the excluded minors for $2$ -regular matroids are $3$ -regular (see [8, Table 5]).

Lemma 4.2.

Let $M$ be an excluded minor for the class of $3$ -regular matroids. Then, either

(i)

$M$ has a $\{U_{2,6},U_{3,6},U_{4,6},P_{6}\}$ -minor, or

(ii)

$M$ is isomorphic to a matroid in

\{F_{7},F_{7}^{*},F_{7}^{-},(F_{7}^{-})^{*},F_{7}^{=},(F_{7}^{=})^{*},\\ \mathit{AG}(2,3)\backslash e,(\mathit{AG}(2,3)\backslash e)^{*},(\mathit{AG}(2,3)\backslash e)^{\Delta Y},P_{8},P_{8}^{-},P_{8}^{=},\mathit{TQ}_{8}\}.

The following is a consequence of the fact that $U_{2,5}$ is a strong $\mathbb{H}_{5}$ -stabilizer [26, Lemma 7.3.16].

Lemma 4.3.

The matroids $U_{2,6}$ , $U_{3,6}$ , $U_{4,6}$ , and $P_{6}$ are strong $\mathbb{H}_{5}$ -stabilizers.

We also require the following consequence of Lemmas 3.2, 3.4 and 4.2.

Lemma 4.4.

If $M$ is a quad-flower or a nest, then $M$ is not an excluded minor for the class of $3$ -regular matroids.

Proof.

If $M$ is a nest, then $M$ has either $F_{7}$ or $F_{7}^{-}$ as a proper minor, by Lemma 3.4, so $M$ is not an excluded minor for the class of $3$ -regular matroids, by Lemma 4.2. Suppose that $M$ is a quad-flower that is an excluded minor for the class of $3$ -regular matroids. Then every proper minor of $M$ is $3$ -regular. By Lemma 3.2, $M$ is either a binary quad-flower, a dyadic quad-flower, or a free quad-flower. But then $M$ has either $F_{7}$ , $F_{7}^{-}$ , or $F_{7}^{=}$ as a proper minor, respectively, contradicting Lemma 4.2. ∎

(a)

F_{7}

(b)

F_{7}^{-}

(c)

F_{7}^{=}

(d)

H_{7}

(e)

M(K_{4})+e

(f)

\mathcal{W}^{3}+e

(g)

\Lambda_{3}

(h)

Q_{6}+e

(i)

R_{6}+e

Figure 2. Nine rank-

3

excluded minors for the class of

3

-regular matroids.

For a matroid $M$ , let $M+e$ denote the matroid obtained from $M$ by a free single-element extension. Consider the set of $10$ matroids that can be obtained from the Fano matroid $F_{7}$ by relaxing circuit-hyperplanes. This gives the sequence

\{F_{7}\},\{F_{7}^{-}\},\{F_{7}^{=}\},\{H_{7},M(K_{4})+e\},\{\mathcal{W}^{3}+e,\Lambda_{3}\},\{Q_{6}+e\},\{P_{6}+e\},\{U_{3,7}\};

geometric representations of these matroids are given in Figure 2. These $10$ matroids give rise to nine $\Delta Y$ -equivalence classes (the matroids $H_{7}$ and $M(K_{4})+e$ are $\Delta Y$ -equivalent). Each of these matroids turns out to be an excluded minor for the class of $3$ -regular matroids.

Theorem 4.5.

There are exactly $33$ matroids that are excluded minors for the class of $3$ -regular matroids and have at most $13$ elements. These are:

F_{7},F_{7}^{-},F_{7}^{=},H_{7},M(K_{4})+e,\mathcal{W}^{3}+e,\Lambda_{3},Q_{6}+e,P_{6}+e,U_{3,7},

and their duals; $\Delta^{(*)}(U_{2,7})$ ; and $\mathit{AG}(2,3)\backslash e$ , $(\mathit{AG}(2,3)\backslash e)^{*}$ , $(\mathit{AG}(2,3)\backslash e)^{\Delta Y}$ , $P_{8}$ , $P_{8}^{-}$ , $P_{8}^{=}$ , and $\mathit{TQ}_{8}$ .

Proof.

Let $\mathcal{U}^{(n)}$ be the set of all $n$ -element $3$ -connected $3$ -regular matroids with a $\{U_{2,6},U_{3,6},U_{4,6},P_{6}\}$ -minor. Using a computer, we exhaustively generated the matroids in $\mathcal{U}^{(n)}$ , for each $n=6,7,8,\dotsc,13$ (see Table 2). By Lemma 4.2, any excluded minor for the class of $3$ -regular matroids has at least $7$ elements. Let $n\in\{7,8,9,10\}$ , and suppose all excluded minors for the class on fewer than $n$ elements are known. We generated all $3$ -connected single-element extensions of some matroid in $\mathcal{U}^{(n-1)}$ ; let $\mathcal{Z}^{(n)}$ be this collection of matroids. From the collection $\mathcal{Z}^{(n)}$ , we filter out any matroids in $\mathcal{U}^{(n)}$ , or any matroid containing, as a minor, one of the excluded minors for $3$ -regular matroids on fewer than $n$ elements; call the resulting collection $\mathcal{Y}^{(n)}$ . Clearly, any matroid in $\mathcal{Y}^{(n)}$ is an excluded minor for the class. In fact, if $M$ is an $n$ -element excluded minor not listed in Lemma 4.2(ii), then, by Lemmas 4.2 and 2.10, the collection $\mathcal{Y}^{(n)}$ contains at least one of $M$ and $M^{*}$ , so the complete list of $n$ -element excluded minors is obtained by closing $\mathcal{Y}^{(n)}$ under duality. Now, all excluded minors for the class on at most $10$ elements are known.

Next consider when $n=11$ . In order to utilise the contrapositive of Lemma 2.15, we consider $3$ -connected matroids with a $\{U_{2,5},U_{3,5}\}$ -minor, so that, when $|E(M)|=11$ and $N\in\{U_{2,5},U_{3,5}\}$ , we have $|E(M)|-|E(N)|=6$ . Let $\mathcal{V}^{(n)}$ be the set of all $n$ -element $3$ -connected $3$ -regular matroids with a $\{U_{2,5},U_{3,5}\}$ -minor. We exhaustively generated the matroids in $\mathcal{V}^{(n)}$ , for each $n\leq 11$ (see Table 3). For any (not necessarily non-isomorphic) pair of matroids $M_{e}$ and $M_{f}$ in $\mathcal{V}^{(10)}$ , where $M_{e}$ and $M_{f}$ are both single-element extensions of some matroid $M^{\prime}\in\mathcal{V}^{(9)}$ , we find the splices of $M_{e}$ and $M_{f}$ . Let $\mathcal{Z}$ be the collection of all $11$ -element matroids that can be obtained in this way. From the collection $\mathcal{Z}$ , we filter out any matroids in $\mathcal{V}^{(11)}$ , or any matroid containing, as a minor, one of the excluded minors for $3$ -regular matroids on at most $10$ elements; call the resulting collection $\mathcal{Y}^{(11)}$ . Clearly, any matroid in $\mathcal{Y}^{(11)}$ is an excluded minor for the class. In fact, if $M$ is an $11$ -element excluded minor, then, by Lemmas 4.2 and 2.14, either the collection $\mathcal{Y}^{(11)}$ contains a matroid that is $\Delta Y$ -equivalent to $M$ or $M^{*}$ , or $M$ has a spiky $3$ -separator. But the latter contradicts Lemma 2.15. Therefore, the complete list of $11$ -element excluded minors is obtained by closing $\mathcal{Y}^{(11)}$ under duality and $\Delta Y$ -equivalence. Now, all excluded minors for the class on at most $11$ elements are known.

For $n\in\{12,13\}$ , the approach is similar to when $n=11$ , only we return to considering $3$ -connected $3$ -regular matroids with a $\{U_{2,6},U_{3,6},U_{4,6},P_{6}\}$ -minor, for efficiency of the computations. Recall that $\mathcal{U}^{(n)}$ is the set of such matroids on $n$ elements, and we have exhaustively generated the matroids in $\mathcal{U}^{(n)}$ for $n\leq 13$ . Let $n\in\{12,13\}$ , and assume we know all excluded minors for the class on $n-1$ elements. For any (not necessarily non-isomorphic) pair of matroids $M_{e}$ and $M_{f}$ in $\mathcal{U}^{(n-1)}$ , where $M_{e}$ and $M_{f}$ are both single-element extensions of some matroid $M^{\prime}\in\mathcal{U}^{(n-2)}$ , we find the splices of $M_{e}$ and $M_{f}$ . Let $\mathcal{Z}^{(n)}$ be the collection of all $n$ -element matroids that can be obtained in this way. We note that $\mathcal{Z}^{(13)}$ contained 7550463 matroids with rank 6, and 6788774 matroids with rank 7. From the collection $\mathcal{Z}^{(n)}$ , we filter out any matroids in $\mathcal{U}^{(n)}$ , or any matroid containing, as a minor, one of the excluded minors for $3$ -regular matroids on at most $n-1$ elements; call the resulting collection $\mathcal{Y}^{(n)}$ . Clearly, any matroid in $\mathcal{Y}^{(n)}$ is an excluded minor for the class. In fact, if $M$ is an $n$ -element excluded minor, then, by Lemmas 4.2, 2.14 and 2.15, either the collection $\mathcal{Y}^{(n)}$ contains a matroid that is $\Delta Y$ -equivalent to either $M$ or $M^{*}$ , or $n=12$ and $M$ is a quad-flower or a nest. But the latter contradicts Lemma 4.4. Therefore, the complete list of $n$ -element excluded minors is obtained by closing $\mathcal{Y}^{(n)}$ under duality and $\Delta Y$ -equivalence. This completes the proof. ∎

$r\backslash n$	6	7	8	9	10	11	12	13
2	1
3	2	2	3	4	3	2	1
4	1	2	15	63	160	252	294	238
5			3	63	572	2969	9701	21766
6				4	160	2969	29288	172233
7					3	252	9701	172233
8						2	294	21766
9							1	238
Total	4	4	21	134	264	6446	49280	388474

Table 2. The number of

3

-connected

3

-regular

n

-element rank-

r

matroids with a

\{U_{2,6},U_{3,6},U_{4,6},P_{6}\}

-minor, for

n\leq 13

$r\backslash n$	5	6	7	8	9	10	11
2	1	1
3	1	3	4	7	7	5	2
4		1	4	32	125	273	384
5				7	125	1074	5125
6					7	273	5125
7						5	384
8							2
9
10
Total	2	5	8	46	264	1630	11022

Table 3. The number of

3

-connected

3

-regular

n

-element rank-

r

matroids with a

\{U_{2,5},U_{3,5}\}

-minor, for

n\leq 11

Comparing with the excluded minors for $2$ -regular matroids, note that instead of $U_{2,6}$ , $U_{3,6}$ , $U_{4,6}$ and $P_{6}$ , the matroids that appear as excluded minors for $3$ -regular matroids are either in $\Delta^{(*)}(U_{2,7})$ , or can be obtained from $F_{7}$ by relaxing zero or more circuit-hyperplanes and then possibly dualising.

In Table 4, we show the excluded minors for $3$ -regular matroids, grouped by $\Delta Y$ -equivalence classes, and, for those that are $\mathrm{GF}(5)$ -representable, we provide the largest integer $i$ for which they are $\mathbb{H}_{i}$ -representable. We also specify their universal partial field, if this partial field is well known.

$M$	$\mathbb{P}_{M}$	$\max\{i:M\in\mathcal{M}(\mathbb{H}_{i})\}$	$\left\|\Delta(M)\right\|$
$U_{2,7}$	$\mathbb{U}_{4}$	–	6
$F_{7}$	$\mathrm{GF}(2)$	–	2
$F_{7}^{-}$	$\mathbb{D}$	2	2
$F_{7}^{=}$	$\mathbb{K}_{2}$	2	2
$M(K_{4})+e$		4	4
$\mathcal{W}^{3}+e$		2	2
$\Lambda_{3}$		4	2
$Q_{6}+e$		2	2
$P_{6}+e$		–	2
$U_{3,7}$		–	2
$\mathit{AG}(2,3)\backslash e$	$\mathbb{S}$	–	3
$P_{8}$	$\mathbb{D}$	2	1
$P_{8}^{-}$	$\mathbb{K}_{2}$	2	1
$P_{8}^{=}$	$\mathbb{H}_{4}$	4	1
$\mathit{TQ}_{8}$	$\mathbb{K}_{2}$	2	1

Table 4. The excluded minors for the class of

3

-regular matroids on at most

13

elements, and their universal partial fields. We list one representative

M

of each

\Delta Y

-equivalence class

\Delta(M)

Recall that the following was proved by Brettell, Oxley, Semple and Whittle [7].

Theorem 4.6 ([7, Theorem 1.1]).

An excluded minor for the class of $3$ -regular matroids has at most $15$ elements.

It is natural to ask the viability of using the techniques here (and in [8]) to find the excluded minors for $3$ -regular matroids on up to $15$ elements, in order to obtain a full excluded-minor characterisation for the class.

For obtaining the excluded minors for the class of $2$ -regular matroids, an important optimisation used in [8] is that any such excluded minor with at least $9$ elements is $\mathrm{GF}(4)$ -representable, for otherwise it would also be an excluded minor for the class of $\mathrm{GF}(4)$ -representable matroids (the largest of which is known to have $8$ elements [14]). Thus, there one can first find certain $\mathrm{GF}(211)$ -representable matroids (as a “proxy” for $2$ -regular matroids), and then the potential excluded minors are $\mathrm{GF}(4)$ -representable extensions (or splices) of these matroids. The same cannot be said for $3$ -regular matroids, where we consider certain $\mathrm{GF}(3527)$ -representable matroids (as a “proxy” for $3$ -regular matroids), and then must consider all extensions (or splices) of these matroids.

The upshot is, using more time or computing resources, one could probably extend Theorem 4.5 to matroids on 14 elements, but then another big step in computational power would still be required to tackle matroids on $15$ elements. A better approach seems some sort of compromise between an exhaustive computation and developing the theory in order to decrease the bound, as in [7]: that is, a computation that exploits structural results.

5. Excluded minors for $\mathbb{H}_{i}$ -representable matroids when $i\in\{1,2,3,4\}$

In this section, we use the following notation. For $i\in\{1,2,3,4,5\}$ , let $\mathcal{X}_{i}$ be the set of excluded minors for the class of $\mathbb{H}_{i}$ -representable matroids. For $i\in\{1,2,3,4\}$ , let $\mathcal{N}_{i}$ be the set of matroids in $\mathcal{X}_{i+1}$ that are $\mathbb{H}_{i}$ -representable. Using this notation, we have:

Lemma 5.1.

Let $M\in\mathcal{X}_{i}$ for some $i\in\{1,2,3,4\}$ . Then either

(i)

$M$ has an $\mathcal{N}_{i}$ -minor, or
(ii)

$M\in\mathcal{X}_{i+1}$ .

Moreover, the matroids in $\mathcal{N}_{i}$ are strong $\mathbb{H}_{i}$ -stabilizers by Lemma 2.7.

We consider now the computation of matroids in $\mathcal{X}_{i}$ on at most $10$ elements, for $i\in\{1,2,3,4\}$ . We first consider when $i=4$ , in which case, using Theorems 4.5 and 4, we can start from the following:

Lemma 5.2.

Suppose $M\in\mathcal{X}_{4}$ and $|E(M)|\leq 13$ . Then, either

(i)

$M$ or $M^{*}$ has a minor in $\{P_{8}^{=},\Lambda_{3},H_{7},M(K_{4})+e\}$ , or

(ii)

$M$ is isomorphic to one of $26$ matroids in $\mathcal{X}_{5}$ ; namely, $M$ or $M^{*}$ is isomorphic to a matroid in

\{F_{7},F_{7}^{-},F_{7}^{=},U_{3,7},\mathcal{W}^{3}+e,Q_{6}+e,P_{6}+e,\\ \mathit{AG}(2,3)\backslash e,(\mathit{AG}(2,3)\backslash e)^{\Delta Y},P_{8},P_{8}^{-},\mathit{TQ}_{8}\}\cup\Delta(U_{2,7}).

Theorem 5.3.

There are $63$ excluded minors for $\mathbb{H}_{4}$ -representability on at most $10$ elements.

Proof.

Let $M\in\mathcal{X}_{4}$ with $|E(M)|\leq 10$ . By Lemma 5.2, if $M$ is not one of the 26 matroids in Lemma 5.2(ii), then $M$ has an $\mathcal{N}_{4}$ -minor, where $\mathcal{N}_{4}$ is obtained by closing $\{P_{8}^{=},\Lambda_{3},H_{7},M(K_{4})+e\}$ under duality.

Let $\mathcal{H}_{4}^{(n)}$ denote the $n$ -element $3$ -connected $\mathbb{H}_{4}$ -representable matroids with an $\mathcal{N}_{4}$ -minor. Starting from each matroid $N\in\mathcal{N}_{4}$ , and using the fact that $N$ is a strong $\mathbb{H}_{4}$ -stabilizer (by Lemma 2.7), we exhaustively generated the matroids in $\mathcal{H}_{4}^{(n)}$ , for $n=7,8,9,10$ , see Table 5. Suppose all matroids in $\mathcal{X}_{4}$ on fewer than $n$ elements are known, for some $n\leq 10$ . We generated all $3$ -connected single-element extensions of some matroid in $\mathcal{H}^{(n-1)}_{4}$ , then removed any matroids in $\mathcal{H}^{(n)}_{4}$ , or any containing, as a minor, an excluded minor for the class of $\mathbb{H}_{4}$ -representable matroids on fewer than $n$ elements. It follows from Lemma 2.10 that, after closing the resulting set of matroids under duality, we obtain all the matroids in $\mathcal{X}_{4}$ having at most $n$ elements. ∎

$r\backslash n$	7	8	9	10
3	3	3	4	3
4	3	24	127	408
5		3	127	1747
6			4	408
7				3
Total	6	30	262	2569

Table 5. The number of

3

-connected

\mathbb{H}_{4}

-representable

n

-element rank-

r

matroids with a

\mathcal{N}_{4}

-minor, for

n\leq 10

For the next layer of the hierarchy, the question becomes: which of the 63 excluded minors are $\mathbb{H}_{3}$ -representable? It turns out that 20 of them are, as given in the next lemma. We provide $\mathbb{H}_{3}$ -representations for these matroids in the appendix.

Note that some of these matroids are related to the well-known Pappus matroid [18, Page 655], which we denote as $\mathsf{Pappus}$ . After deleting an element from $\mathsf{Pappus}$ , the resulting matroid is unique up to isomorphism; we denote this matroid as $\mathsf{Pappus}\backslash e$ . The matroid $\mathsf{Pappus}\backslash e$ has a pair of free bases (see [10, Lemma 2.1]); tightening either one of these results in a matroid that we denote $(\mathsf{Pappus}\backslash e)^{+}$ (the two possible matroids are isomorphic).

Lemma 5.4.

If $N\in\mathcal{N}_{3}$ and $E(N)\leq 10$ , then $N$ is one of the $20$ matroids obtained by closing the set

\Delta(\{\mathsf{Pappus}\backslash e,(\mathsf{Pappus}\backslash e)^{+},M_{3240}\})\cup\{M_{1543},M_{1559},M_{1566},M_{1596},M_{1598},Fq\}

under duality.

Note that, by Theorems 5.3 and 5.4, there are 43 excluded minors for $\mathbb{H}_{4}$ -representability that are also excluded minors for $\mathbb{H}_{3}$ -representability.

Theorem 5.5.

There are $133$ excluded minors for $\mathbb{H}_{3}$ -representability on at most $10$ elements.

Proof.

Essentially the same approach was taken as for the proof of Theorem 5.3, only using Lemma 5.4. The number of $n$ -element $3$ -connected $\mathbb{H}_{3}$ -representable matroids with an $\mathcal{N}_{3}$ -minor are given in Table 6. ∎

$r\backslash n$	8	9	10
3	2	3	1
4	9	65	231
5	2	65	1144
6		3	231
7			1
Total	13	136	1608

Table 6. The number of

3

-connected

\mathbb{H}_{3}

-representable

n

-element rank-

r

matroids with a

\mathcal{N}_{3}

-minor, for

n\leq 10

Lemma 5.6.

If $N\in\mathcal{N}_{2}$ and $E(N)\leq 10$ , then $N$ is one of the $99$ matroids obtained by closing the set

\{F_{7}^{-},F_{7}^{=},\mathcal{W}^{3}+e,Q_{6}+e,P_{8},P_{8}^{-},\mathit{TQ}_{8},\\ M_{1537},M_{1554},M_{1556},M_{1569},M_{1676},\mathit{Fr}_{1},\mathit{Fr}_{2},\mathit{Fr}_{3},\\ M_{3241},M_{10782},M_{16834},M_{17078},M_{17759},M_{47015},\mathit{Fs}_{1},\mathit{Fs}_{2},\mathit{Fs}_{3},\mathit{Fs}_{4},\\ \mathit{Ft}_{1},\mathit{Ft}_{2},\mathit{Ft}_{4},\mathit{Ft}_{5},\mathit{Ft}_{6},\mathit{Ft}_{7},\mathit{Ft}_{8},\mathit{Fu}_{3},\mathit{Fu}_{5},\mathit{Fu}_{8}\}

under $\Delta Y$ -equivalence and duality.

Note that, by Theorems 5.5 and 5.6, there are $34$ excluded minors for $\mathbb{H}_{3}$ -representability that are also excluded minors for $\mathbb{H}_{2}$ -representability.

Theorem 5.7.

There are $621$ excluded minors for $\mathbb{H}_{2}$ -representability on at most $10$ elements.

Proof.

Again, we use a similar approach as in the proof of Theorems 5.5 and 5.3, only using Lemma 5.6. The number of $n$ -element $3$ -connected $\mathbb{H}_{2}$ -representable matroids with an $\mathcal{N}_{2}$ -minor are given in Table 7. ∎

$r\backslash n$	7	8	9	10
3	4	13	32	37
4	4	89	913	5125
5		13	913	21839
6			32	5125
7				37
Total	8	115	1890	32163

Table 7. The number of

3

-connected

\mathbb{H}_{2}

-representable

n

-element rank-

r

matroids with a

\mathcal{N}_{2}

-minor, for

n\leq 10

Lemma 5.8.

If $N\in\mathcal{N}_{1}$ and $E(N)\leq 10$ , then $N$ is one of the $441$ matroids obtained by closing the set

\{M_{1601},\lambda_{4}^{+},\mathit{Ft}_{3},\mathit{Fu}_{1},\mathit{Fu}_{6},\mathit{Fu}_{7},\mathit{UG}_{10},M_{822},M_{835},M_{836},M_{854},M_{1495},M_{1505},\\ M_{1507},\mathit{KP}_{8},M_{1519},M_{1530},M_{1535},M_{1538},M_{1541},M_{1542},M_{1553},M_{1591},M_{1593},M_{1599},\\ M_{1600},M_{1609},M_{1610},M_{1618},M_{1627},M_{1674},M_{1680},M_{1695},M_{1698},M_{1713},M_{1729},\\ M_{3193},\mathit{BB}_{9},A_{9},\mathit{BB}_{9}^{\Delta\nabla},M_{9574},M_{9841},M_{9844},\mathit{PP}_{9},M_{10025},M_{10052},M_{10241},M_{10242},\\ M_{10244},M_{10251},M_{10287},M_{10368},M_{10459},M_{11161},M_{11162},M_{11347},M_{12000},M_{12575},\\ M_{15729},\mathit{TQ}_{9},M_{16321},M_{16332},M_{18663},M_{46942},M_{46945},M_{47635},M_{48819},M_{50041},\\ \mathit{Fv}_{6},\mathit{Fv}_{7},\mathit{Fv}_{8},\mathit{Fv}_{11},\mathit{Fv}_{12},\mathit{Fv}_{16},\mathit{Fv}_{18},\mathit{Fv}_{19},\mathit{Fv}_{21},\mathit{Fv}_{24},\mathit{Fv}_{25},\mathit{Fv}_{27},\mathit{Fv}_{37},\mathit{Fv}_{41},\\ \mathit{Fv}_{42},\mathit{Fv}_{47},\mathit{Fv}_{51},\mathit{Fv}_{54},\mathit{Fv}_{56},\mathit{Fv}_{60},\mathit{Fv}_{61},\mathit{Fv}_{66},\mathit{Fv}_{68},\mathit{Fv}_{71},\mathit{Fv}_{75},\mathit{Fv}_{77},\mathit{Fv}_{85},\mathit{Fv}_{87},\\ \mathit{Fv}_{90},\mathit{Fv}_{91},\mathit{Fv}_{103},\mathit{GP}_{10},\mathit{TQ}_{10},\mathit{FF}_{10},\mathit{Fv}_{14},\mathit{Fv}_{22},\mathit{Fv}_{26},\mathit{Fv}_{28},\mathit{Fv}_{30},\mathit{Fv}_{32},\mathit{Fv}_{35},\\ \mathit{Fv}_{36},\mathit{Fv}_{39},\mathit{Fv}_{40},\mathit{Fv}_{43},\mathit{Fv}_{44},\mathit{Fv}_{46},\mathit{Fv}_{48},\mathit{Fv}_{49},\mathit{Fv}_{50},\mathit{Fv}_{53},\mathit{Fv}_{55},\mathit{Fv}_{57},\mathit{Fv}_{58},\mathit{Fv}_{59},\\ \mathit{Fv}_{62},\mathit{Fv}_{63},\mathit{Fv}_{65},\mathit{Fv}_{70},\mathit{Fv}_{72},\mathit{Fv}_{73},\mathit{Fv}_{78},\mathit{Fv}_{79},\mathit{Fv}_{81},\mathit{Fv}_{82},\mathit{Fv}_{83},\mathit{Fv}_{84},\mathit{Fv}_{89},\mathit{Fv}_{92},\\ \mathit{Fv}_{93},\mathit{Fv}_{94},\mathit{Fv}_{95},\mathit{Fv}_{96},\mathit{Fv}_{101},\mathit{Fv}_{102},\mathit{Fv}_{104},\mathit{Fv}_{106},\mathit{Fv}_{111},\mathit{Fv}_{112}\}

under $\Delta Y$ -equivalence and duality.

Let $M$ be a uniquely $\mathrm{GF}(5)$ -representable matroid that is not dyadic. Then $M$ has an $\mathcal{N}_{1}$ -minor. In particular, if $|E(M)|\leq 10$ , then $M$ or $M^{*}$ is $\Delta Y$ -equivalent to a matroid that has an $N$ -minor, for some matroid $N$ listed in Lemma 5.8.

Note that, by Theorems 5.7 and 5.8, there are $180$ excluded minors for $\mathbb{H}_{2}$ -representability that are also excluded minors for the class of $\mathrm{GF}(5)$ -representable matroids.

Finally, we obtain our main result, Theorem 1.1, which we restate here.

Theorem 1.1.

There are $2128$ excluded minors for $\mathrm{GF}(5)$ -representability on $10$ elements.

Proof.

Again, we use a similar approach as in the proof of Theorems 5.3, 5.5 and 5.7, only this time using Lemma 5.8. The number of $n$ -element rank- $r$ $3$ -connected $\mathrm{GF}(5)$ -representable matroids with an $\mathcal{N}_{1}$ -minor are given in Table 8. ∎

$r\backslash n$	8	9	10
3	4	33	120
4	46	2237	51823
5	4	2237	327186
6		33	51823
7			120
Total	54	4540	431072

Table 8. The number of

3

-connected

n

-element rank-

r

matroids that are uniquely

\mathrm{GF}(5)

-representable but not dyadic, for

n\leq 10

6. Final remarks

In Table 9, we provide the number of excluded minors for $\mathrm{GF}(5)$ -representability of each rank and size (up to at most 10 elements). We also consider the number of $\Delta Y$ -equivalence classes for each size.

As remarked prior to Proposition 2.8, the excluded minors for representability over a partial field $\mathbb{P}$ are closed not only under $\Delta$ - $Y$ exchange, but also under segment-cosegment exchange (a generalisation of $\Delta$ - $Y$ exchange) [19]. Thus, we could instead consider the number of equivalence classes under segment-cosegment exchange; however, the only effect would be merging two of the classes on $10$ elements, reducing the number of equivalence classes from $920$ to $919$ .

$n$	$r$	#(matroids)	#( $\Delta Y^{(*)}$ -classes)
7	2 / 5	1
	3 / 4	5
	Total	12	4
8	3 / 5	2
	4	92
	Total	96	73
9	3 / 6	9
	4 / 5	219
	Total	456	131
10	3 / 7	1
	4 / 6	130
	5	1866
	Total	2128	920

Table 9. The number of excluded minors for the class of

\mathrm{GF}(5)

-representable matroids on

n

elements and rank

r

, and the number of

\Delta Y

-equivalence classes.

Of the $920$ dual-closed $\Delta Y$ -equivalence classes of $10$ -element excluded minors for the class of $\mathrm{GF}(5)$ -representable matroids, $513$ of the classes consist of matroids that are not representable over any field. This corresponds to $1235$ out of the $2128$ excluded minors on $10$ elements.

A partial field $\mathbb{P}$ is universal if it is the universal partial field of some matroid $M$ ; that is, $\mathbb{P}=\mathbb{P}_{M}$ . Van Zwam showed that several well-known partial fields are universal [26, Table 3.2], but commented that it was not known if each Hydra- $i$ partial field is universal, for $i\in\{2,3,4,5\}$ [26, page 92]. It follows from Lemma 2.5 that $\mathbb{P}_{U_{2,6}}=\mathbb{H}_{5}$ . In the process of finding the excluded minors for $\mathrm{GF}(5)$ , we were also able to find a matroid $M$ such that $\mathbb{P}_{M}=\mathbb{H}_{i}$ for each $i$ , thereby showing that each Hydra partial field is universal. These are given in Table 10. We note that each matroid in this table has at most $9$ elements.

$\mathbb{P}_{M}$	$\mathbb{H}_{5}$	$\mathbb{H}_{4}$	$\mathbb{H}_{3}$	$\mathbb{H}_{2}$	$\mathrm{GF}(5)$
$M$	$U_{2,6}$	$P_{8}^{=}$	$(\mathsf{Pappus}\backslash e)^{+}$	$M_{16834}$	$M_{15729}$

Table 10. Matroids with Hydra universal partial fields

References

[1] S. Akkari and J. Oxley. Some local extremal connectivity results for matroids. Combinatorics, Probability and Computing, 2(04):367–384, 1993.
[2] M. Baker and O. Lorscheid. Lift theorems for representations of matroids over pastures. arXiv:2107.00981, 2021.
[3] R. E. Bixby. On Reid’s characterization of the ternary matroids. Journal of Combinatorial Theory, Series B, 26(2):174–204, 1979.
[4] J. E. Bonin and W. R. Schmitt. Splicing matroids. European Journal of Combinatorics, 32:722–744.
[5] N. Brettell. On the excluded minors for the class of matroids representable over all fields of size at least four. In preparation.
[6] N. Brettell, B. Clark, J. Oxley, C. Semple, and G. Whittle. Excluded minors are almost fragile. Journal of Combinatorial Theory, Series B, 140:263–322, 2020.
[7] N. Brettell, J. Oxley, C. Semple, and G. Whittle. The excluded minors for $2$ - and $3$ -regular matroids. Journal of Combinatorial Theory, Series B, 163:133–218, 2023.
[8] N. Brettell and R. Pendavingh. Computing excluded minors for classes of matroids representable over partial fields. The Electronic Journal of Combinatorics, 31(P3.20):1–23, 2024.
[9] N. Brettell, G. Whittle, and A. Williams. $N$ -detachable pairs in $3$ -connected matroids III: the theorem. Journal of Combinatorial Theory, Series B, 153:223–290, 2022.
[10] R. Chen and G. Whittle. On recognizing frame and lifted-graphic matroids. Journal of Graph Theory, 87(1):72–76, 2018.
[11] A. L. C. Cheung. Compatibility of extensions of a combinatorial geometry. Ph.D., University of Waterloo, 1974.
[12] J. Geelen, B. Gerards, and G. Whittle. Solving Rota’s conjecture. Notices of the AMS, 61(7):736–743, 2014.
[13] J. Geelen, J. Oxley, D. Vertigan, and G. Whittle. Weak maps and stabilizers of classes of matroids. Advances in Applied Mathematics, (21):305–341, 1998.
[14] J. F. Geelen, A. M. H. Gerards, and A. Kapoor. The excluded minors for GF $(4)$ -representable matroids. Journal of Combinatorial Theory, Series B, 79(2):247–299, 2000.
[15] R. Hall, D. Mayhew, and S. H. M. van Zwam. The excluded minors for near-regular matroids. European Journal of Combinatorics, 32(6):802–830, 2011.
[16] D. Mayhew and G. F. Royle. Matroids with nine elements. Journal of Combinatorial Theory, Series B, 98(2):415–431, 2008.
[17] W. Myrvold and J. Woodcock. A large set of torus obstructions and how they were discovered. The Electronic Journal of Combinatorics, 25(1.16):1–17, 2018.
[18] J. Oxley. Matroid Theory, volume 21 of Oxford Graduate Texts in Mathematics. Oxford University Press, New York, second edition, 2011.
[19] J. Oxley, C. Semple, and D. Vertigan. Generalized $\Delta$ - $Y$ exchange and $k$ -regular matroids. Journal of Combinatorial Theory, Series B, 79(1):1–65, 2000.
[20] J. Oxley, D. Vertigan, and G. Whittle. On inequivalent representations of matroids over finite fields. Journal of Combinatorial Theory, Series B, 67:325–343, 1996.
[21] R. A. Pendavingh and S. H. M. van Zwam. Confinement of matroid representations to subsets of partial fields. Journal of Combinatorial Theory, Series B, 100(6):510–545, 2010.
[22] R. A. Pendavingh and S. H. M. van Zwam. Lifts of matroid representations over partial fields. Journal of Combinatorial Theory, Series B, 100(1):36–67, 2010.
[23] C. A. Semple. $k$ -Regular Matroids. Ph.D., Victoria University of Wellington, 1998.
[24] P. D. Seymour. Matroid representation over GF $(3)$ . Journal of Combinatorial Theory, Series B, 26(2):159–173, 1979.
[25] W. T. Tutte. A homotopy theorem for matroids. I, II. Transactions of the American Mathematical Society, 88(1):144–174, 1958.
[26] S. H. M. van Zwam. Partial Fields in Matroid Theory. Ph.D., Eindhoven University of Technology, 2009.
[27] G. Whittle. On matroids representable over GF $(3)$ and other fields. Transactions of the American Mathematical Society, 349(2):579–603, 1997.
[28] Y. Yu. More forbidden minors for wye-delta-wye reducibility. The Electronic Journal of Combinatorics, 13(R7):1–15, 2006.

Appendix

Here we provide representations for the $\mathrm{GF}(5)$ -representable matroids appearing as excluded minors for $\mathbb{H}_{i}$ -representable matroids, for some $i\in\{2,3,4,5\}$ .

In what follows, matroids appear just once. That is, when $N$ is a excluded minor for the class of $\mathbb{H}_{i}$ -representable matroids, and $N$ is $\mathbb{H}_{j}$ -representable but not $\mathbb{H}_{j+1}$ -representable for some $j<i-1$ , then $N$ appears in the “Excluded minors for $\mathbb{H}_{i}$ -representable matroids” section under the “ $\mathbb{H}_{j}$ -representable” heading (but not in the “Excluded minors for $\mathbb{H}_{i-1}$ -representable matroids” section, for example, despite $N$ also being an excluded minor for this class).

Matroids on at most nine elements are typically given the name $M_{n}$ , where $n$ is the number in Mayhew and Royle’s catalog of all matroids on at most nine elements. Exceptions are made for well-known matroids that have notation associated with them, or are free extensions or single-element deletions of such matroids, or for matroids that have been discovered in ongoing work on matroids representable over all fields of size at least four [5].

Excluded minors for $\mathbb{H}_{5}$ -representable matroids

•

$\mathbb{H}_{4}$ -representable:

M(K_{4})+e

\begin{bmatrix}1&\alpha&\alpha&1\\ 0&1&1&1\\ 1&0&\alpha&\frac{\beta(\alpha-1)}{1-\beta}\\ \end{bmatrix}

\Lambda_{3}

\begin{bmatrix}1&1&\alpha+\beta-2\alpha\beta&\alpha\beta-1\\ 1&\alpha&0&\alpha(\beta-1)\\ 1&0&\alpha(1-\beta)&\alpha(\beta-1)\\ \end{bmatrix}

P_{8}^{=}

\begin{bmatrix}1&-1&-1&1\\ 1&\frac{\alpha+\beta-2\alpha\beta}{\alpha(\beta-1)}&0&1\\ 1&-1&\frac{(\alpha-1)(\beta-1)}{\alpha\beta-1}&0\\ \frac{1}{\beta-1}&1&1&\frac{\alpha(\beta-1)}{\alpha+\beta-2\alpha\beta}\\ \end{bmatrix}

•

$\mathbb{H}_{2}$ -representable:

F_{7}^{-}

\begin{bmatrix}0&1&1&1\\ 1&0&1&1\\ 1&1&0&1\end{bmatrix}

F_{7}^{=}

\begin{bmatrix}0&1&i&1\\ 1&0&1&1\\ 1&1&0&1\end{bmatrix}

\mathcal{W}^{3}+e

\begin{bmatrix}1&0&i&1\\ i&1&0&1\\ 0&i&1&1\end{bmatrix}

Q_{6}+e

\begin{bmatrix}\frac{i+1}{2}&0&i&1\\ 1&1&1&1\\ 0&\frac{1-i}{2}&-i&1\end{bmatrix}

P_{8}

\begin{bmatrix}0&1&1&2\\ 1&0&1&1\\ 1&1&0&1\\ 2&1&1&0\\ \end{bmatrix}

P_{8}^{-}

\begin{bmatrix}1&1&1&i+1\\ 1&0&i+1&i+1\\ 1&-i&1&0\\ 0&1&1&1\\ \end{bmatrix}

\mathit{TQ}_{8}

\begin{bmatrix}0&i&1&1\\ 1&0&i&i-1\\ 1&i&0&i\\ 1&i-1&1&0\\ \end{bmatrix}

Excluded minors for $\mathbb{H}_{4}$ -representable matroids

•

$\mathbb{H}_{3}$ -representable:

\mathsf{Pappus}\backslash e

\begin{bmatrix}1&1&0&\alpha&\frac{\alpha^{2}}{\alpha-1}\\ 1&0&\alpha&\alpha-1&\alpha\\ 0&1&1-\alpha&1&1\\ \end{bmatrix}

(\mathsf{Pappus}\backslash e)^{+}

\begin{bmatrix}1&1&0&\alpha&\alpha\\ 1&0&\alpha&\alpha-1&\alpha\\ 0&1&1-\alpha&1&1\\ \end{bmatrix}

M_{1559}

\begin{bmatrix}\alpha(1-\alpha)&1&1&1\\ 1&\frac{\alpha-1}{\alpha^{2}}&1&1\\ \alpha&0&\alpha&1\\ 0&1&\alpha&1\\ \end{bmatrix}

M_{1596}

\begin{bmatrix}1-\alpha&\alpha&1&1\\ \frac{-\alpha}{1-\alpha}&\frac{1}{1-\alpha}&1&1\\ 1&0&\alpha&1\\ 0&\alpha&\alpha&1\\ \end{bmatrix}

M_{1543}

\begin{bmatrix}1&1&1&1\\ \alpha&\alpha&0&1\\ \alpha&0&\alpha^{2}&\alpha^{2}-\alpha+1\\ 0&1&1-\alpha&1-\alpha\\ \end{bmatrix}

M_{1566}

\begin{bmatrix}1&1&\frac{\alpha-1}{\alpha}&1\\ 1&\alpha&\alpha&1\\ \frac{\alpha}{\alpha-1}&0&1&1-\alpha\\ 0&1&1&1\\ \end{bmatrix}

M_{1598}

\begin{bmatrix}1&\alpha&1&1\\ 1&\alpha-1&\alpha&\alpha\\ \frac{\alpha}{\alpha^{2}-\alpha+1}&0&\alpha&1\\ 0&\alpha&\alpha&1\\ \end{bmatrix}

M_{3240}

\begin{bmatrix}1&0&\frac{\alpha^{2}}{\alpha^{2}-\alpha+1}&\frac{\alpha^{2}}{\alpha^{2}-\alpha+1}&1&\alpha\\ 1&\alpha^{2}&\alpha&0&\alpha&\alpha^{2}\\ 1&\alpha-1&1&1&1&\alpha-1\\ \end{bmatrix}

\mathit{Fq}

\begin{bmatrix}1&1&1&1&0\\ 0&\alpha(\alpha-1)&0&\alpha^{2}-\alpha+1&\alpha^{2}-\alpha+1\\ \alpha&0&\alpha&1&0\\ \alpha^{2}&\alpha&0&\alpha-1&\alpha-1\\ \alpha(1-\alpha)&0&\alpha&1&1\\ \end{bmatrix}

•

$\mathbb{H}_{2}$ -representable:

M_{1537}

\begin{bmatrix}1&1&1&1\\ \frac{1}{i}&\frac{1}{i}&1&1\\ -1&0&i&1\\ 0&1&i&1\end{bmatrix}

M_{1554}

\begin{bmatrix}1&1&1&1\\ -1&-1&1&1\\ i&0&i&1\\ 0&-i&i&1\end{bmatrix}

M_{1556}

\begin{bmatrix}1&1&1&1\\ -\frac{1}{i}&-\frac{1}{i}&1&1\\ -1&0&i&1\\ 0&1&i&1\end{bmatrix}

M_{1569}

\begin{bmatrix}1&1&2&1\\ 1&1-i&1-i&1\\ i&0&i(1-i)&1\\ 0&1&1&1\end{bmatrix}

M_{1676}

\begin{bmatrix}1&1&1&1\\ -1&-1&1&1\\ i&0&-1&1\\ 0&-i&-1&1\end{bmatrix}

\mathit{Fr}_{1}

\begin{bmatrix}1&i&1&0&0\\ 1&1&1&1&1\\ 1&i&1&i&i\\ 1&-i&0&1&0\\ 1&0&0&1&\frac{i+1}{2}\\ \end{bmatrix}

\mathit{Fr}_{2}

\begin{bmatrix}1&1&1&1&1\\ 1&i+1&0&0&1-i\\ 1&0&\frac{1-i}{2}&1&1\\ 1&0&1-i&2&0\\ 1&i+1&0&2&i+1\\ \end{bmatrix}

\mathit{Fr}_{3}

\begin{bmatrix}-i&1&0&1&-i\\ 0&1&\frac{i+1}{2}&1&0\\ i&1&i&0&0\\ 1&1&1&0&1\\ 0&1&0&1&\frac{1-i}{2}\\ \end{bmatrix}

•

Uniquely $\mathrm{GF}(5)$ -representable:

M_{1601}

\begin{bmatrix}1&1&1&1\\ 1&4&3&0\\ 1&1&4&4\\ 1&4&0&3\\ \end{bmatrix}

\lambda_{4}^{+}

\begin{bmatrix}1&1&1&1\\ 1&3&0&4\\ 1&3&1&0\\ 1&2&3&3\\ \end{bmatrix}

Excluded minors for $\mathbb{H}_{3}$ -representable matroids

•

$\mathbb{H}_{2}$ -representable:

M_{3241}

\begin{bmatrix}\frac{2}{1+i}&\frac{2}{1-i}&0&1&1&\frac{2}{1-i}\\ 1&0&\frac{1-i}{2}&1&1&1\\ 0&1&1&\frac{1}{1-i}&1&1\\ \end{bmatrix}

M_{10782}

\begin{bmatrix}0&i&-i&0&1\\ 0&0&1&\frac{1-i}{2}&1\\ \frac{i}{1+i}&1&0&0&1\\ 1&1&1&1&1\\ \end{bmatrix}

M_{16834}

\begin{bmatrix}1&1&1&1&0\\ 1&1-i&1-i&0&0\\ 1&0&2&1-i&1\\ 1&1-i&0&1&i\\ \end{bmatrix}

M_{17078}

\begin{bmatrix}1&1&i+1&\frac{i+1}{2}&0\\ 1&1&i&0&-i\\ \frac{i+1}{2}&1&0&\frac{1}{2}&1\\ 0&1&1&1&1\\ \end{bmatrix}

M_{17759}

\begin{bmatrix}0&i&1&1&i+1\\ 1&1&1&1&1\\ 0&-1&0&1&i\\ 1&0&\frac{i}{i+1}&1&i\\ \end{bmatrix}

M_{47015}

\begin{bmatrix}1&1&0&\frac{1-i}{2}&\frac{1+i}{2}\\ 1&1&1&1&1\\ 1&i+1&i&1&i\\ 1&i+1&i&0&i+1\\ \end{bmatrix}

\mathit{Fs}_{1}

\begin{bmatrix}\frac{i}{i+1}&1&1&0&i&1&\frac{1}{i+1}\\ \frac{i+1}{2}&1&1&1&0&\frac{i+1}{2}&1\\ \frac{1}{i+1}&\frac{1}{i+1}&1&1&1&\frac{1}{i+1}&\frac{1}{i+1}\\ \end{bmatrix}

\mathit{Fs}_{2}

\begin{bmatrix}0&-i&0&i&-i&1\\ 1&1&1&1&1&1\\ i&i&0&i&1&1\\ i&0&\frac{i+1}{2}&i&1&1\\ \end{bmatrix}

\mathit{Fs}_{3}

\begin{bmatrix}0&\frac{i+1}{2}&1&\frac{1-i}{2}&0&0\\ 1&1&1&0&i&-1\\ 1&0&1&1&1&1\\ \frac{i+1}{i}&0&0&1&i+1&0\\ \end{bmatrix}

\mathit{Fs}_{4}

\begin{bmatrix}0&0&1&\frac{i+1}{2}&\frac{1-i}{2}&0\\ 1&1&1&1&1&1\\ \frac{i+1}{2}&1&1&\frac{i+1}{2}&\frac{1-i}{2}&\frac{1-i}{2}\\ 1&1-i&1&0&1-i&-i\\ \end{bmatrix}

\mathit{Ft}_{1}

\begin{bmatrix}i+1&\frac{i+1}{2}&1&i+1&0\\ 1&1&1&1&1\\ 1&1&1&0&0\\ i+1&\frac{i+1}{2}&1&i&1\\ 0&0&1&i&1\\ \end{bmatrix}

\mathit{Ft}_{2}

\begin{bmatrix}1&0&1-i&i+1&2\\ 1&1&0&1-i&0\\ 1&i+1&i+1&1-i&0\\ 1&1&1&1&1\\ 1&0&0&1&\frac{2}{i+1}\\ \end{bmatrix}

\mathit{Ft}_{4}

\begin{bmatrix}1&i&0&\frac{i+1}{2}&i+1\\ 1&1&1&1&1\\ 1&i&0&i&i\\ 1&-i&1&0&0\\ 1&1&1&\frac{i+1}{2}&0\\ \end{bmatrix}

\mathit{Ft}_{5}

\begin{bmatrix}i+1&i&i+1&1&0\\ i+1&0&\frac{i+1}{2}&1&1\\ 1&0&0&1&1\\ 1&1&1&1&1\\ 0&1&0&1&i+1\\ \end{bmatrix}

\mathit{Ft}_{6}

\begin{bmatrix}1&1&1&1&1\\ 1&i&i&0&0\\ 1&0&1&1-i&1\\ 1&0&1&0&\frac{1}{i}\\ 1&i&i&1&1\\ \end{bmatrix}

\mathit{Ft}_{7}

\begin{bmatrix}1&1&1&1&1\\ 2&1&1&2&1\\ \frac{2}{i+1}&1&0&2&1\\ 1&1&1&i+1&\frac{i+1}{2}\\ 0&1&0&i+1&i\\ \end{bmatrix}

\mathit{Ft}_{8}

\begin{bmatrix}1&1&1&1&1\\ 1&i+1&1-i&0&0\\ 1&0&1&-i&\frac{1-i}{2}\\ 1&0&1&0&1\\ 1&1-i&0&1&1\\ \end{bmatrix}

\mathit{Fu}_{3}

\begin{bmatrix}0&1&1&i&1\\ 0&1&1&-1&0\\ \frac{1}{2}&0&1&0&\frac{1}{2}\\ 1&1&1&1&1\\ \frac{i+1}{2}&0&1&1&1\\ \end{bmatrix}

\mathit{Fu}_{5}

\begin{bmatrix}1&0&0&\frac{1-i}{2}&\frac{1-i}{2}\\ 1&1&1&1&1\\ 1&\frac{1}{2}&0&\frac{1}{2}&\frac{1}{2}\\ 1&1&1&0&\frac{1}{2}\\ 1&\frac{i+1}{2}&1&\frac{1-i}{2}&\frac{1}{2}\\ \end{bmatrix}

\mathit{Fu}_{8}

\begin{bmatrix}0&-1&1&i&1\\ 1&1&1&0&i\\ 1&1&1&1&1\\ \frac{1-i}{2}&0&1&0&1\\ 0&i&1&\frac{1+i}{2}&0\\ \end{bmatrix}

•

Uniquely $\mathrm{GF}(5)$ -representable:

\mathit{Ft}_{3}

\begin{bmatrix}1&1&1&1&1\\ 1&2&2&0&0\\ 1&0&1&4&2\\ 1&0&1&0&1\\ 1&2&2&1&1\\ \end{bmatrix}

\mathit{Fu}_{1}

\begin{bmatrix}1&1&1&1&1\\ 1&3&3&0&0\\ 1&0&1&3&1\\ 1&0&1&0&3\\ 1&3&3&1&1\\ \end{bmatrix}

\mathit{Fu}_{6}

\begin{bmatrix}1&0&4&1&1\\ 1&1&0&4&1\\ 1&4&3&0&0\\ 1&1&1&1&1\\ 1&1&0&2&4\\ \end{bmatrix}

\mathit{Fu}_{7}

\begin{bmatrix}0&1&1&1&1\\ 1&0&0&1&1\\ 1&1&3&0&0\\ 0&1&4&2&4\\ 2&1&0&4&1\\ \end{bmatrix}

\mathit{UG}_{10}

\begin{bmatrix}0&2&1&4&1\\ 1&1&1&0&4\\ 1&1&1&1&1\\ 3&0&1&0&1\\ 0&4&1&2&0\\ \end{bmatrix}

Excluded minors for $\mathbb{H}_{2}$ -representable matroids

•

Uniquely $\mathrm{GF}(5)$ -representable:

M_{822}

\begin{bmatrix}0&3&1&1&1\\ 4&0&1&1&2\\ 1&1&1&0&1\\ \end{bmatrix}

M_{835}

\begin{bmatrix}1&1&0&1&2\\ 2&0&1&1&1\\ 0&3&2&1&1\\ \end{bmatrix}

M_{836}

\begin{bmatrix}3&0&1&1&2\\ 0&1&1&2&4\\ 1&1&1&1&1\\ \end{bmatrix}

M_{854}

\begin{bmatrix}4&0&2&1&1\\ 0&2&3&3&1\\ 1&1&1&1&1\\ \end{bmatrix}

M_{1495}

\begin{bmatrix}1&1&2&0\\ 1&0&2&2\\ 1&1&1&1\\ 1&2&0&2\\ \end{bmatrix}

M_{1505}

\begin{bmatrix}3&1&2&0\\ 1&1&1&1\\ 0&1&2&2\\ 2&1&0&2\\ \end{bmatrix}

M_{1507}

\begin{bmatrix}4&1&1&0\\ 1&1&1&1\\ 0&1&2&2\\ 2&1&0&2\\ \end{bmatrix}

\mathit{KP}_{8}

\begin{bmatrix}1&3&3&0\\ 1&1&1&1\\ 1&3&2&2\\ 1&4&0&1\\ \end{bmatrix}

M_{1519}

\begin{bmatrix}1&4&4&0\\ 1&0&4&1\\ 1&2&3&2\\ 1&1&1&1\\ \end{bmatrix}

M_{1530}

\begin{bmatrix}1&4&2&0\\ 1&0&2&2\\ 1&1&1&1\\ 1&2&0&2\\ \end{bmatrix}

M_{1535}

\begin{bmatrix}1&1&4&0\\ 1&4&3&3\\ 1&1&1&1\\ 1&3&0&3\\ \end{bmatrix}

M_{1538}

\begin{bmatrix}1&2&3&0\\ 1&2&4&4\\ 1&1&1&1\\ 1&4&0&4\\ \end{bmatrix}

M_{1541}

\begin{bmatrix}2&2&1&0\\ 2&0&1&1\\ 0&2&1&3\\ 1&1&1&1\\ \end{bmatrix}

M_{1542}

\begin{bmatrix}1&1&1&1\\ 1&4&0&2\\ 1&4&4&0\\ 1&2&4&1\\ \end{bmatrix}

M_{1553}

\begin{bmatrix}1&0&3&1\\ 1&4&0&4\\ 1&4&1&0\\ 1&1&1&1\\ \end{bmatrix}

M_{1591}

\begin{bmatrix}3&0&1&3\\ 0&4&1&2\\ 3&1&1&0\\ 1&1&1&1\\ \end{bmatrix}

M_{1593}

\begin{bmatrix}1&1&1&1\\ 1&4&2&1\\ 1&3&4&3\\ 1&2&3&3\\ \end{bmatrix}

M_{1599}

\begin{bmatrix}1&1&1&1\\ 1&3&4&3\\ 1&2&1&4\\ 1&4&3&0\\ \end{bmatrix}

M_{1600}

\begin{bmatrix}1&1&1&1\\ 1&4&1&2\\ 1&1&4&0\\ 1&2&4&1\\ \end{bmatrix}

M_{1609}

\begin{bmatrix}1&1&1&1\\ 1&3&3&4\\ 1&4&3&0\\ 1&2&0&1\\ \end{bmatrix}

M_{1610}

\begin{bmatrix}1&1&1&1\\ 1&3&0&4\\ 1&4&2&0\\ 1&2&2&1\\ \end{bmatrix}

M_{1618}

\begin{bmatrix}1&1&1&1\\ 1&1&4&3\\ 1&3&4&0\\ 1&2&0&1\\ \end{bmatrix}

M_{1627}

\begin{bmatrix}1&1&1&1\\ 1&2&1&4\\ 1&1&3&0\\ 1&2&3&1\\ \end{bmatrix}

M_{1674}

\begin{bmatrix}1&1&1&1\\ 1&0&3&4\\ 1&4&3&0\\ 1&2&4&1\\ \end{bmatrix}

M_{1680}

\begin{bmatrix}1&1&1&1\\ 1&4&4&3\\ 1&3&1&0\\ 1&2&4&1\\ \end{bmatrix}

M_{1695}

\begin{bmatrix}1&1&1&1\\ 1&4&0&2\\ 1&2&1&0\\ 1&4&2&1\\ \end{bmatrix}

M_{1698}

\begin{bmatrix}1&1&1&1\\ 1&3&3&2\\ 1&2&3&0\\ 1&4&2&1\\ \end{bmatrix}

M_{1713}

\begin{bmatrix}1&0&4&1\\ 1&2&4&3\\ 1&2&1&0\\ 1&1&1&1\\ \end{bmatrix}

M_{1729}

\begin{bmatrix}1&1&1&1\\ 0&1&2&4\\ 1&1&2&0\\ 2&1&4&2\\ \end{bmatrix}

M_{3193}

\begin{bmatrix}1&0&1&0&3&1\\ 1&4&0&1&0&1\\ 0&1&1&1&1&1\\ \end{bmatrix}

\mathit{BB}_{9}

\begin{bmatrix}1&0&1&4&1&4\\ 1&3&1&3&0&0\\ 0&1&1&1&1&1\\ \end{bmatrix}

A_{9}

\begin{bmatrix}1&3&3&4&0&4\\ 1&3&2&3&3&0\\ 1&1&1&1&1&1\\ \end{bmatrix}

\mathit{BB}_{9}^{\Delta\nabla}

\begin{bmatrix}3&0&1&0\\ 0&3&1&1\\ 1&1&1&1\\ 1&2&0&3\\ 1&0&4&0\\ \end{bmatrix}

M_{9574}

\begin{bmatrix}2&0&1&1&1\\ 0&3&1&4&0\\ 1&1&1&0&2\\ 0&1&1&2&0\\ \end{bmatrix}

M_{9841}

\begin{bmatrix}3&0&1&1&0\\ 1&1&1&1&1\\ 0&2&1&2&0\\ 1&2&1&0&2\\ \end{bmatrix}

M_{9844}

\begin{bmatrix}2&0&1&2&0\\ 1&1&1&1&1\\ 0&2&1&1&0\\ 1&2&1&2&1\\ \end{bmatrix}

\mathit{PP}_{9}

\begin{bmatrix}1&1&1&4&4\\ 1&2&1&0&4\\ 1&0&4&4&4\\ 0&1&1&1&1\\ \end{bmatrix}

M_{10025}

\begin{bmatrix}4&0&1&4&0\\ 1&1&1&1&1\\ 0&4&1&2&0\\ 1&2&1&2&2\\ \end{bmatrix}

M_{10052}

\begin{bmatrix}4&0&1&4&0\\ 1&1&1&1&1\\ 0&2&1&1&0\\ 1&2&1&2&1\\ \end{bmatrix}

M_{10241}

\begin{bmatrix}4&0&1&3&4\\ 1&2&1&2&0\\ 0&4&1&1&0\\ 1&1&1&1&1\\ \end{bmatrix}

M_{10242}

\begin{bmatrix}3&0&1&3&0\\ 1&1&1&1&1\\ 0&2&1&4&2\\ 1&4&1&4&0\\ \end{bmatrix}

M_{10244}

\begin{bmatrix}4&0&3&1&1\\ 4&1&4&1&1\\ 0&1&2&2&1\\ 1&1&1&1&1\\ \end{bmatrix}

M_{10251}

\begin{bmatrix}3&0&1&4&3\\ 1&4&1&2&4\\ 0&2&1&1&2\\ 1&1&1&1&1\\ \end{bmatrix}

M_{10287}

\begin{bmatrix}2&0&1&3&0\\ 1&4&1&0&0\\ 0&1&1&2&2\\ 1&1&1&1&1\\ \end{bmatrix}

M_{10368}

\begin{bmatrix}4&0&1&2&1\\ 0&2&1&3&0\\ 1&1&1&0&4\\ 0&1&1&1&1\\ \end{bmatrix}

M_{10459}

\begin{bmatrix}4&0&1&4&0\\ 1&1&1&1&1\\ 0&2&1&1&2\\ 1&2&1&2&0\\ \end{bmatrix}

M_{11161}

\begin{bmatrix}2&0&1&1&0\\ 1&1&1&1&1\\ 0&3&1&2&0\\ 1&2&1&0&2\\ \end{bmatrix}

M_{11162}

\begin{bmatrix}3&0&1&1&0\\ 1&1&1&1&1\\ 0&3&1&4&0\\ 1&4&1&0&4\\ \end{bmatrix}

M_{11347}

\begin{bmatrix}4&0&1&3&0\\ 1&4&1&0&0\\ 0&1&1&4&4\\ 1&1&1&1&1\\ \end{bmatrix}

M_{12000}

\begin{bmatrix}3&0&1&2&4\\ 1&1&1&1&1\\ 0&3&1&4&0\\ 1&4&1&0&3\\ \end{bmatrix}

M_{12575}

\begin{bmatrix}2&0&1&1&0\\ 1&1&1&1&1\\ 0&4&1&3&0\\ 1&3&1&0&3\\ \end{bmatrix}

M_{15729}

\begin{bmatrix}3&0&1&2&2\\ 1&3&1&4&0\\ 0&1&1&3&2\\ 1&1&1&1&1\\ \end{bmatrix}

\mathit{TQ}_{9}

\begin{bmatrix}1&0&2&1&1\\ 4&0&0&2&1\\ 1&4&0&0&2\\ 1&1&1&1&0\\ \end{bmatrix}

M_{16321}

\begin{bmatrix}3&0&1&2&3\\ 1&2&1&0&1\\ 0&1&1&3&2\\ 1&1&1&1&1\\ \end{bmatrix}

M_{16332}

\begin{bmatrix}3&0&1&1&1\\ 1&3&1&2&0\\ 0&1&1&4&1\\ 1&1&1&3&0\\ \end{bmatrix}

M_{18663}

\begin{bmatrix}2&0&1&1&1\\ 1&4&1&3&0\\ 0&1&1&2&1\\ 1&1&1&4&0\\ \end{bmatrix}

M_{46942}

\begin{bmatrix}3&0&1&3&1\\ 1&1&1&1&1\\ 0&1&3&3&1\\ 3&1&3&1&0\\ \end{bmatrix}

M_{46945}

\begin{bmatrix}4&0&1&4&2\\ 1&1&1&1&1\\ 0&2&1&1&0\\ 1&2&1&2&2\\ \end{bmatrix}

M_{47635}

\begin{bmatrix}3&0&1&0&1\\ 1&1&1&1&1\\ 0&2&1&3&2\\ 1&4&1&3&2\\ \end{bmatrix}

M_{48819}

\begin{bmatrix}4&0&1&2&2\\ 1&4&1&0&1\\ 0&1&1&4&2\\ 1&1&1&1&1\\ \end{bmatrix}

M_{50041}

\begin{bmatrix}3&0&1&1&4\\ 1&1&1&1&1\\ 0&3&1&4&3\\ 1&4&1&0&1\\ \end{bmatrix}

\mathit{Fv}_{6}

\begin{bmatrix}1&4&0&1&1&2\\ 1&4&0&1&3&1\\ 1&2&4&3&2&4\\ 1&1&1&1&1&1\\ \end{bmatrix}

\mathit{Fv}_{7}

\begin{bmatrix}1&1&1&1&0&3\\ 2&1&1&0&1&1\\ 4&0&1&0&1&1\\ 0&2&1&1&0&3\\ \end{bmatrix}

\mathit{Fv}_{8}

\begin{bmatrix}4&1&0&1&2&1\\ 1&1&2&1&2&3\\ 1&1&1&1&1&1\\ 0&0&3&1&1&1\\ \end{bmatrix}

\mathit{Fv}_{11}

\begin{bmatrix}1&3&1&1&1&3\\ 2&0&1&1&3&1\\ 0&1&1&2&1&2\\ 3&2&1&0&3&2\\ \end{bmatrix}

\mathit{Fv}_{12}

\begin{bmatrix}1&1&1&3&4&0\\ 1&3&2&2&0&3\\ 1&3&2&0&1&2\\ 1&1&1&1&1&1\\ \end{bmatrix}

\mathit{Fv}_{16}

\begin{bmatrix}1&0&1&1&1\\ 1&0&3&0&2\\ 1&1&0&3&0\\ 4&1&4&0&1\\ 0&1&0&2&4\\ \end{bmatrix}

\mathit{Fv}_{18}

\begin{bmatrix}0&3&1&1&1\\ 0&2&3&0&1\\ 2&4&2&3&1\\ 4&0&4&0&1\\ 1&1&1&1&1\\ \end{bmatrix}

\mathit{Fv}_{19}

\begin{bmatrix}1&1&1&1&1\\ 0&2&1&0&1\\ 3&2&1&2&2\\ 2&0&1&0&4\\ 0&2&1&1&3\\ \end{bmatrix}

\mathit{Fv}_{21}

\begin{bmatrix}4&0&1&0&1\\ 0&0&1&4&4\\ 1&1&1&1&1\\ 1&4&1&4&4\\ 1&3&1&0&1\\ \end{bmatrix}

\mathit{Fv}_{24}

\begin{bmatrix}1&1&1&1&1\\ 1&3&2&1&3\\ 1&0&3&3&0\\ 1&1&2&3&0\\ 1&3&2&0&2\\ \end{bmatrix}

\mathit{Fv}_{25}

\begin{bmatrix}1&1&1&1&1\\ 1&1&2&0&1\\ 2&1&4&1&1\\ 0&2&0&2&1\\ 2&2&1&1&1\\ \end{bmatrix}

\mathit{Fv}_{27}

\begin{bmatrix}1&1&1&1&1\\ 0&3&1&0&4\\ 0&3&1&3&0\\ 3&0&1&0&1\\ 2&3&1&1&1\\ \end{bmatrix}

\mathit{Fv}_{37}

\begin{bmatrix}1&1&1&1&1\\ 0&2&1&0&4\\ 3&2&1&2&3\\ 2&0&1&0&2\\ 0&2&1&1&1\\ \end{bmatrix}

\mathit{Fv}_{41}

\begin{bmatrix}1&1&1&1&1\\ 3&1&4&4&1\\ 3&1&4&2&2\\ 0&1&2&1&1\\ 0&1&0&2&4\\ \end{bmatrix}

\mathit{Fv}_{42}

\begin{bmatrix}0&0&2&1&1\\ 4&0&0&1&1\\ 1&1&1&1&1\\ 1&1&2&1&3\\ 0&4&1&1&3\\ \end{bmatrix}

\mathit{Fv}_{47}

\begin{bmatrix}1&1&1&1&1\\ 1&4&1&4&4\\ 1&1&0&3&3\\ 1&0&0&1&3\\ 1&1&1&4&0\\ \end{bmatrix}

\mathit{Fv}_{51}

\begin{bmatrix}0&1&2&1&1\\ 1&1&1&1&1\\ 4&2&0&1&1\\ 0&4&3&1&2\\ 1&0&1&1&0\\ \end{bmatrix}

\mathit{Fv}_{54}

\begin{bmatrix}0&2&1&1&0\\ 0&2&1&0&4\\ 1&1&1&3&0\\ 4&0&1&0&4\\ 3&0&1&1&1\\ \end{bmatrix}

\mathit{Fv}_{56}

\begin{bmatrix}1&1&1&1&1\\ 0&1&1&2&1\\ 1&1&4&1&2\\ 2&1&3&1&3\\ 2&1&3&3&0\\ \end{bmatrix}

\mathit{Fv}_{60}

\begin{bmatrix}2&2&0&1&1\\ 4&4&3&1&0\\ 3&0&1&1&0\\ 1&1&1&1&1\\ 0&3&0&1&1\\ \end{bmatrix}

\mathit{Fv}_{61}

\begin{bmatrix}2&2&0&1&0\\ 4&4&3&1&0\\ 3&0&1&1&1\\ 1&1&1&1&1\\ 0&3&0&1&1\\ \end{bmatrix}

\mathit{Fv}_{66}

\begin{bmatrix}1&1&0&3&3\\ 0&1&1&1&1\\ 1&1&0&1&0\\ 3&1&4&0&0\\ 0&1&1&4&3\\ \end{bmatrix}

\mathit{Fv}_{68}

\begin{bmatrix}1&0&1&1&1\\ 1&0&4&0&2\\ 1&4&0&4&0\\ 2&1&2&0&1\\ 0&1&0&2&1\\ \end{bmatrix}

\mathit{Fv}_{71}

\begin{bmatrix}0&2&1&2&2\\ 0&2&1&0&4\\ 1&1&1&1&1\\ 3&0&1&0&4\\ 4&0&1&2&3\\ \end{bmatrix}

\mathit{Fv}_{75}

\begin{bmatrix}1&1&0&3&0\\ 0&1&1&1&1\\ 1&1&0&1&1\\ 3&1&4&0&0\\ 0&1&1&4&2\\ \end{bmatrix}

\mathit{Fv}_{77}

\begin{bmatrix}1&1&1&1&0\\ 0&2&1&0&2\\ 3&2&1&2&0\\ 2&0&1&0&2\\ 0&2&1&1&1\\ \end{bmatrix}

\mathit{Fv}_{85}

\begin{bmatrix}4&3&1&3&2\\ 0&1&1&0&2\\ 1&1&1&1&1\\ 1&0&1&0&4\\ 1&1&1&3&2\\ \end{bmatrix}

\mathit{Fv}_{87}

\begin{bmatrix}1&3&0&1&1\\ 1&1&1&1&1\\ 1&1&3&2&4\\ 1&1&2&2&2\\ 1&4&0&2&3\\ \end{bmatrix}

\mathit{Fv}_{90}

\begin{bmatrix}0&2&1&2&0\\ 0&1&1&0&4\\ 1&1&1&1&1\\ 1&0&1&0&2\\ 1&1&1&2&3\\ \end{bmatrix}

\mathit{Fv}_{91}

\begin{bmatrix}0&4&1&1&3\\ 2&0&1&2&0\\ 1&1&1&1&1\\ 1&1&1&3&2\\ 4&2&1&2&0\\ \end{bmatrix}

\mathit{Fv}_{103}

\begin{bmatrix}1&1&1&1&1\\ 1&0&4&1&1\\ 1&0&4&3&3\\ 1&2&1&3&1\\ 1&1&1&0&4\\ \end{bmatrix}

\mathit{GP}_{10}

\begin{bmatrix}4&3&0&1&1\\ 3&3&0&0&1\\ 0&0&1&2&1\\ 1&0&2&0&1\\ 1&1&1&1&0\\ \end{bmatrix}

\mathit{TQ}_{10}

\begin{bmatrix}3&0&0&1&4\\ 0&0&1&2&2\\ 0&1&1&1&1\\ 1&1&1&0&4\\ 4&1&2&4&4\\ \end{bmatrix}

\mathit{FF}_{10}

\begin{bmatrix}1&0&0&1&1\\ 1&1&0&0&1\\ 1&3&1&0&0\\ 1&1&1&2&0\\ 3&3&3&2&2\\ \end{bmatrix}

\mathit{Fv}_{14}

\begin{bmatrix}1&1&1&1&1\\ 2&1&1&0&1\\ 2&0&0&1&1\\ 0&0&1&1&1\\ 0&4&2&1&0\\ \end{bmatrix}

\mathit{Fv}_{22}

\begin{bmatrix}4&0&3&0&1\\ 1&1&1&1&1\\ 4&0&0&2&1\\ 0&1&2&0&1\\ 2&4&0&2&1\\ \end{bmatrix}

\mathit{Fv}_{26}

\begin{bmatrix}0&1&0&2&1\\ 0&0&2&3&1\\ 2&0&0&2&1\\ 1&1&1&1&1\\ 4&1&1&0&1\\ \end{bmatrix}

\mathit{Fv}_{28}

\begin{bmatrix}0&0&1&2&3\\ 1&1&1&1&1\\ 1&3&1&2&1\\ 1&1&1&3&4\\ 3&1&1&2&1\\ \end{bmatrix}

\mathit{Fv}_{30}

\begin{bmatrix}1&1&1&1&1\\ 0&2&0&1&1\\ 1&1&3&1&2\\ 2&1&0&1&3\\ 0&0&1&1&1\\ \end{bmatrix}

\mathit{Fv}_{32}

\begin{bmatrix}2&0&0&1&2\\ 1&1&1&1&1\\ 3&3&1&1&1\\ 0&2&1&1&0\\ 4&4&0&1&1\\ \end{bmatrix}

\mathit{Fv}_{35}

\begin{bmatrix}1&2&2&1&2\\ 1&1&1&1&1\\ 1&0&2&3&0\\ 1&1&3&1&2\\ 1&4&1&0&0\\ \end{bmatrix}

\mathit{Fv}_{36}

\begin{bmatrix}4&4&1&0&1\\ 1&1&1&1&0\\ 3&3&0&1&1\\ 0&4&2&0&1\\ 1&0&0&1&1\\ \end{bmatrix}

\mathit{Fv}_{39}

\begin{bmatrix}0&3&1&3&2\\ 0&2&1&0&3\\ 1&1&1&1&1\\ 1&0&1&0&4\\ 1&4&1&3&2\\ \end{bmatrix}

\mathit{Fv}_{40}

\begin{bmatrix}0&0&1&2&1\\ 4&0&0&2&1\\ 1&1&1&1&1\\ 2&2&1&2&1\\ 0&2&4&4&1\\ \end{bmatrix}

\mathit{Fv}_{43}

\begin{bmatrix}0&0&1&2&1\\ 2&1&3&3&1\\ 1&1&1&1&1\\ 3&4&2&1&1\\ 2&3&1&1&1\\ \end{bmatrix}

\mathit{Fv}_{44}

\begin{bmatrix}0&3&1&3&2\\ 0&2&1&0&3\\ 1&1&1&1&1\\ 1&0&1&0&2\\ 1&4&1&3&2\\ \end{bmatrix}

\mathit{Fv}_{46}

\begin{bmatrix}1&1&1&1&1\\ 1&4&1&3&4\\ 4&4&1&4&0\\ 0&3&1&3&1\\ 2&4&1&1&1\\ \end{bmatrix}

\mathit{Fv}_{48}

\begin{bmatrix}3&2&1&2&0\\ 0&1&1&0&1\\ 1&1&1&1&1\\ 2&0&1&0&2\\ 0&1&1&2&0\\ \end{bmatrix}

\mathit{Fv}_{49}

\begin{bmatrix}3&3&4&3&1\\ 0&2&0&3&1\\ 1&1&1&1&1\\ 3&2&0&2&1\\ 0&0&4&3&1\\ \end{bmatrix}

\mathit{Fv}_{50}

\begin{bmatrix}1&4&3&3&1\\ 0&0&1&4&1\\ 1&1&1&1&1\\ 3&2&4&1&1\\ 4&4&1&1&1\\ \end{bmatrix}

\mathit{Fv}_{53}

\begin{bmatrix}1&1&3&1&2\\ 3&2&0&1&1\\ 1&1&1&1&1\\ 0&1&0&1&2\\ 3&4&3&1&1\\ \end{bmatrix}

\mathit{Fv}_{55}

\begin{bmatrix}2&0&3&1&0\\ 1&1&1&1&1\\ 3&3&3&1&0\\ 0&2&0&1&1\\ 4&4&3&1&1\\ \end{bmatrix}

\mathit{Fv}_{57}

\begin{bmatrix}0&1&1&4&3\\ 2&2&1&0&0\\ 1&1&1&1&1\\ 2&2&1&4&3\\ 0&4&1&1&1\\ \end{bmatrix}

\mathit{Fv}_{58}

\begin{bmatrix}1&1&1&1&1\\ 4&0&1&4&4\\ 0&2&1&2&0\\ 3&3&1&1&1\\ 0&0&1&4&3\\ \end{bmatrix}

\mathit{Fv}_{59}

\begin{bmatrix}3&0&1&0&1\\ 0&0&1&1&1\\ 1&2&1&2&0\\ 1&1&1&1&1\\ 1&3&1&0&2\\ \end{bmatrix}

\mathit{Fv}_{62}

\begin{bmatrix}1&1&1&1&1\\ 0&2&0&4&1\\ 4&4&1&4&1\\ 2&4&0&4&1\\ 0&0&1&1&1\\ \end{bmatrix}

\mathit{Fv}_{63}

\begin{bmatrix}0&4&1&2&2\\ 2&0&1&4&0\\ 1&1&1&2&0\\ 1&1&1&1&1\\ 4&2&1&4&1\\ \end{bmatrix}

\mathit{Fv}_{65}

\begin{bmatrix}0&0&1&3&1\\ 1&1&1&2&3\\ 1&2&1&3&3\\ 1&1&1&1&1\\ 2&1&1&3&3\\ \end{bmatrix}

\mathit{Fv}_{70}

\begin{bmatrix}1&3&1&3&1\\ 0&1&1&0&4\\ 1&1&1&1&1\\ 4&0&1&0&1\\ 0&1&1&3&0\\ \end{bmatrix}

\mathit{Fv}_{72}

\begin{bmatrix}2&3&1&3&1\\ 0&1&1&0&3\\ 1&1&1&1&1\\ 4&0&1&0&3\\ 0&1&1&3&1\\ \end{bmatrix}

\mathit{Fv}_{73}

\begin{bmatrix}4&0&1&0&2\\ 1&1&1&1&1\\ 4&0&1&3&0\\ 0&2&1&0&1\\ 3&1&1&3&1\\ \end{bmatrix}

\mathit{Fv}_{78}

\begin{bmatrix}4&1&0&1&1\\ 0&0&1&3&1\\ 1&1&1&1&1\\ 3&3&1&3&1\\ 1&2&0&1&1\\ \end{bmatrix}

\mathit{Fv}_{79}

\begin{bmatrix}4&0&1&0&3\\ 0&0&1&4&3\\ 1&1&1&1&1\\ 1&4&1&4&3\\ 1&3&1&0&2\\ \end{bmatrix}

\mathit{Fv}_{81}

\begin{bmatrix}0&1&0&1&1\\ 0&1&1&0&1\\ 1&1&1&1&0\\ 3&0&4&0&1\\ 4&0&0&1&1\\ \end{bmatrix}

\mathit{Fv}_{82}

\begin{bmatrix}4&0&1&2&3\\ 2&1&1&2&0\\ 1&1&1&1&0\\ 0&4&1&1&1\\ 0&4&1&4&3\\ \end{bmatrix}

\mathit{Fv}_{83}

\begin{bmatrix}4&0&1&0&4\\ 1&1&1&1&1\\ 4&0&1&2&0\\ 0&4&1&0&1\\ 2&1&1&2&1\\ \end{bmatrix}

\mathit{Fv}_{84}

\begin{bmatrix}1&1&1&3&3\\ 0&4&1&0&3\\ 1&1&1&1&1\\ 2&0&1&0&3\\ 0&0&1&1&1\\ \end{bmatrix}

\mathit{Fv}_{89}

\begin{bmatrix}1&1&1&1&1\\ 0&4&1&0&1\\ 2&4&1&4&0\\ 2&0&1&0&4\\ 2&4&1&1&1\\ \end{bmatrix}

\mathit{Fv}_{92}

\begin{bmatrix}1&2&1&2&1\\ 0&1&1&0&2\\ 1&1&1&1&1\\ 2&0&1&0&1\\ 0&1&1&2&0\\ \end{bmatrix}

\mathit{Fv}_{93}

\begin{bmatrix}1&1&1&2&2\\ 0&2&1&0&2\\ 1&1&1&1&1\\ 4&0&1&0&2\\ 0&0&1&1&1\\ \end{bmatrix}

\mathit{Fv}_{94}

\begin{bmatrix}2&3&0&1&1\\ 0&0&3&1&1\\ 0&2&1&0&1\\ 4&1&0&1&1\\ 1&0&2&0&1\\ \end{bmatrix}

\mathit{Fv}_{95}

\begin{bmatrix}1&1&1&1&1\\ 1&1&1&0&0\\ 4&2&1&2&0\\ 4&0&1&0&4\\ 4&4&1&1&1\\ \end{bmatrix}

\mathit{Fv}_{96}

\begin{bmatrix}1&1&1&1&1\\ 1&1&1&0&0\\ 1&3&2&3&0\\ 1&0&2&0&3\\ 1&1&2&2&2\\ \end{bmatrix}

\mathit{Fv}_{101}

\begin{bmatrix}2&3&1&1&1\\ 1&0&2&1&2\\ 0&4&0&1&1\\ 1&1&1&1&1\\ 1&0&0&1&3\\ \end{bmatrix}

\mathit{Fv}_{102}

\begin{bmatrix}3&1&0&1&3\\ 1&1&1&1&1\\ 0&0&1&1&4\\ 3&3&2&1&3\\ 1&2&1&1&3\\ \end{bmatrix}

\mathit{Fv}_{104}

\begin{bmatrix}1&1&4&0&1\\ 0&0&1&1&1\\ 3&2&0&1&0\\ 1&1&1&0&4\\ 4&1&0&2&4\\ \end{bmatrix}

\mathit{Fv}_{106}

\begin{bmatrix}2&3&1&3&0\\ 0&1&1&0&1\\ 1&1&1&1&1\\ 4&0&1&0&4\\ 0&1&1&3&0\\ \end{bmatrix}

\mathit{Fv}_{111}

\begin{bmatrix}1&1&1&1&1\\ 0&2&1&0&4\\ 0&2&1&2&3\\ 3&0&1&0&1\\ 4&2&1&1&1\\ \end{bmatrix}

\mathit{Fv}_{112}

\begin{bmatrix}1&1&1&1&1\\ 2&1&1&2&2\\ 1&0&1&3&1\\ 4&2&1&4&0\\ 2&4&1&0&1\\ \end{bmatrix}

The excluded minors for GF​(5)\mathrm{GF}(5)-representable matroids on ten elements

Abstract.

1. Introduction

Theorem 1.1.

Theorem 1.2.

2. Preliminaries

Partial fields

Lemma 2.1 ([21, Theorem 2.8]).

Lemma 2.2 ([21, Corollary 2.9]).

Stabilizers

The Hydra-ii partial fields

Lemma 2.3 ([21, Theorem 5.7]).

Corollary 2.4 (see [21, Lemma 5.17]).

The kk-regular partial fields

Lemma 2.5 ([7, Lemma 2.25]).

Corollary 2.6.

Lemma 2.7 ([26, Lemmas 6.2.5 and 7.3.15]).

Other partial fields

Delta-wye exchange

Proposition 2.8 ([19, Theorem 1.1]).

Proxies

Lemma 2.9.

Splitter theorems

Lemma 2.10 ([8, Corollary 2.8]).

Definition 2.11.

Definition 2.12.

Definition 2.13.

Theorem 2.14 ([9, Corollary 6.3]).

Lemma 2.15 (see [6, Lemma 7.2]).

3. Exceptional 1212-element matroids

Quad-flowers

Lemma 3.1.

Proof.

Lemma 3.2.

Proof.

Nests

Lemma 3.3.

Lemma 3.4.

Proof.

4. Excluded minors for 33-regular matroids

Theorem 4.1 ([7, Theorem 1.2]).

Lemma 4.2.

Lemma 4.3.

Lemma 4.4.

Proof.

Theorem 4.5.

Proof.

Theorem 4.6 ([7, Theorem 1.1]).

5. Excluded minors for ℍi\mathbb{H}_{i}-representable matroids when i∈{1,2,3,4}i\in\{1,2,3,4\}

Lemma 5.1.

Lemma 5.2.

Theorem 5.3.

Proof.

Lemma 5.4.

Theorem 5.5.

Proof.

Lemma 5.6.

Theorem 5.7.

Proof.

Lemma 5.8.

Theorem 1.1.

Proof.

6. Final remarks

References

Appendix

Excluded minors for ℍ5\mathbb{H}_{5}-representable matroids

Excluded minors for ℍ4\mathbb{H}_{4}-representable matroids

Excluded minors for ℍ3\mathbb{H}_{3}-representable matroids

Excluded minors for ℍ2\mathbb{H}_{2}-representable matroids

The excluded minors for $\mathrm{GF}(5)$ -representable matroids on ten elements

The Hydra- $i$ partial fields

The $k$ -regular partial fields

3. Exceptional $12$ -element matroids

4. Excluded minors for $3$ -regular matroids

5. Excluded minors for $\mathbb{H}_{i}$ -representable matroids when $i\in\{1,2,3,4\}$

Excluded minors for $\mathbb{H}_{5}$ -representable matroids

Excluded minors for $\mathbb{H}_{4}$ -representable matroids

Excluded minors for $\mathbb{H}_{3}$ -representable matroids

Excluded minors for $\mathbb{H}_{2}$ -representable matroids