Partial fields give a more flexible theory of matroid representation than mere fields. Rather than committing to a field, a partial field allows you to capture several representations at once. They were first introduced by Semple and Whittle [SW96]. The archetypical example is the following theorem by Tutte:

**Theorem 1.** *The following statements are equivalent for a matroid $M$:
*

*$M$ is representable over both $\text{GF}(2)$ and $\text{GF}(3)$;**$M$ is representable over $\text{GF}(2)$ and some field of characteristic other than $2$;**$M$ is representable over $\mathbb{R}$ by a totally unimodular matrix;*- $M$
*is representable over*every*field.*

Matroid representations over partial fields generalize totally unimodular matrices, as we will see below. We know that regular matroids (i.e. the matroids from Theorem 1) possess many interesting properties. Let $\mathcal{M}$ be the class of regular matroids.

- The excluded minors for $\mathcal{M}$ are $U_{2,4}$, $F_7$, and $F_7^*$.
- Every regular matroid is uniquely representable over every field (up to row operations and column scaling).
- All regular matroids can be obtained from graphic matroids and $R_{10}$ by dualizing and 1-, 2-, 3-sums (this is
*Seymour’s Decomposition Theorem*). - A simple, regular matroid of rank $r$ has at most ${r+1 \choose 2}$ elements, with equality for the graphic matroids of complete graphs.
- If $A$ is a totally unimodular matrix of full row rank representing $M$, then $\det(A A^T)$ equals the number of bases of $M$.

For every partial field $\mathbb{P}$, one may ask if analogues of these properties exist for the class of $\mathbb{P}$-representable matroids. This is an area of research with a number of interesting results and many unsolved problems.

In this post I will limit myself to introducing matroid representation over partial fields, linking them to the definition of totally unimodular matrices, and giving some examples. In my next post I will discuss Whittle’s characterization [Whi95, 97] of the representations of ternary matroids.

# 1. Definitions

The definition of a partial field is straightforward:

**Definition 2. **A *partial field* is a pair $\mathbb{P} = (R, G)$, where $R$ is a commutative ring, and $G$ is a subgroup of the invertible elements of $R$ such that $-1 \in G$.

I will introduce matroid representations over partial fields through *chain groups*. For fields, this corresponds to the row space of the representation matrix. The advantage is a very clean basic theory, plus an extension to noncommutative rings at no extra cost. Incidentally, this is how Tutte himself thought about regular matroids.

**Definition 3. **Let $R$ be a ring, and $E$ a finite set. A *chain group* is a subset $C \subseteq R^E$ such that, for $c, d \in C$ and $r \in R$,

- $0 \in C$;
- $c+d \in C$;
- $rc \in C$.

The elements of a chain group are called *chains*.

**Definition 4. **The *support* of a chain $c\in C$ is

$$\| c \| := \{e \in E : c_e \neq 0\}.$$

A nonzero chain with inclusionwise minimal support is called *elementary.*

**Definition 5. **Let $G \subseteq R$. A chain $c \in R^E$ is *$G$-primitive* if $c \in (G\cup\{0\})^E$.

**Definition 6. **Let $\mathbb{P} = (R, G)$ be a partial field, and $E$ a finite set. A chain group $C \subseteq R^E$ is a *$\mathbb{P}$-chain group *if, for every elementary chain $c$ there exists an $r\in R$ such that $c = rg$ for some $G$-primitive chain $g$.

I realize this was a fairly long string of definitions, but the payoff is that we can very quickly define matroid representations:

**Theorem 7. ***Let $\mathbb{P}$ be a partial field, and $C$ a $\mathbb{P}$-chain group. Define*

* $$\mathcal{C}^* := \{ \|c\| : c \in C, \text{ elementary}\}.$$*

* Then $\mathcal{C}^*$ is the collection of cocircuits of a matroid, $M(C)$.*

*Proof.* We verify the (co)circuit axioms.

*$\emptyset\not\in \mathcal{C}^*$.*

An elementary chain was defined to be nonzero.*If $C, D \in \mathcal{C}^*$ and $C \subseteq D$ then $C = D$.*

The supports of elementary chains are inclusionwise minimal.*If $C, D \in \mathcal{C}^*$ are distinct, and $e \in C \cap D$ then there is a member $F \in \mathcal{C}^*$ with $F \subseteq (C\cup D) – \{e\}$.*

Let $c, d$ be chains with $\|c\| = C$ and $\|d\| = D$. Since these chains are elementary, we may assume they are $G$-primitive, i.e. every nonzero entry is invertible. Now write

$$g := c_e^{-1} c – d_e^{-1} d.$$

Then $g$ is nonzero, and therefore there is an elementary chain $f$ whose support is contained in that of $g$. We simply take $F = \|f\|$. $\square$

**Definition 8. **We say a matroid $M$ is $\mathbb{P}$-representable if there exists a $\mathbb{P}$-chain group $C$ such that $M = M(C)$.

One thing to note is that everything above holds even if we don’t assume $R$ to be commutative. This theory of *skew partial fields* was explored in [PZ13]. For the remainder of this post we’ll stick to the commutative case, though.

# 2. Restrictions on determinants

If you’ve seen totally unimodular matrices defined, you’ve probably seen a definition involving determinants. This also works for partial fields:

**Definition 9. **Let $\mathbb{P} = (R, G)$ be a partial field. A *$\mathbb{P}$-matrix *is a matrix $A$ with entries in $R$ such that each square submatrix has a determinant in $G \cup \{0\}$.

We denote by $A[X]$ the submatrix of $A$ with columns indexed by $X$.

**Theorem 10. ***Let $\mathbb{P} = (R, G)$ be a partial field, and $A$ a $\mathbb{P}$-matrix with $r$ rows and columns indexed by $E$. Define*

* $$\mathcal{B}_A := \{ X \subseteq E : |X| = r \text{ and } \det(A[X]) \neq 0\}.$$*

* If this set is nonempty, then $\mathcal{B}_A$ is the set of bases of a matroid, $M[A]$.*

*Proof. *We use a trick from commutative algebra. Let $I$ be a maximal ideal of $R$. Such an ideal is guaranteed to exist (assuming the axiom of choice). Then $\mathbb{F} := R/I$ is a field. Consider the ring homomorphism $\phi: R \to \mathbb{F}$ sending $r$ to $r + I$. Since this is a homomorphism, it commutes with addition and multiplication. Hence, if we apply it to the entries of $A$, we preserve the nonzero determinants. Since we have a matroid after applying $\phi$, we must have had a matroid before. $\square$

The link with the previous section is simple:

**Theorem 11. ***Let $C$ be the chain group generated by the rows of a $\mathbb{P}$-matrix $A$. Then $C$ is a $\mathbb{P}$-chain group and $M(C) = M[A]$.*

We also have a converse:

**Theorem 12. ***Let $C$ be a $\mathbb{P}$-chain group, let $B$ be a basis of $M(C)$, and let $A$ be the matrix whose rows are $G$-primitive chains such that their supports form the $B$-fundamental cocircuits of $M(C)$. Then $A$ is a $\mathbb{P}$-matrix, and $M[A] = M(C)$.*

This post is already getting long, so I will skip the proof and refer to [PZ13, Section 3.4] instead.

# 3. Examples

The key takeaway from the proof of Theorem 10 is the idea of a *homomorphism*:

**Definition 13. **Let $\mathbb{P}_1 = (R_1, G_1)$ and $\mathbb{P}_2 = (R_2, G_2)$ be partial fields, and let $\phi: R_1 \to R_2$ be a ring homomorphism such that $\phi(G_1) \subseteq G_2$. Then $\phi$ is a *partial field homomorphism.*

**Theorem 14. **

*If $\phi$ is as above, then every $\mathbb{P}_1$-representable matroid is also $\mathbb{P}_2$-representable.*

We omit the proof, which is only a slight modification of the proof of Theorem 10. Note that we can identify a field $\mathbb{F}$ with the partial field $(\mathbb{F}, \mathbb{F}^*)$.

The *regular* partial field is $\mathbb{U}_0 = (\mathbb{Z}, \{-1, 1\})$. The $\mathbb{U}_0$-matrices are precisely the totally unimodular matrices. There is clearly a partial field homomorphism to *any *partial field, so regular matroids are, in fact, representable over every *partial** *field too!

The *dyadic *partial field is $\mathbb{D} = (\mathbb{Z}[\frac{1}{2}], \langle -1, 2 \rangle)$. This means that we extended the ring of integers with all powers of $\frac{1}{2}$, and consider matrices where all subdeterminants are, in absolute value, a (positive or negative) power of $2$. There is a ring homomorphism to every field of characteristic other than 2.

The *golden ratio *partial field is $\mathbb{G} = (\mathbb{Z}[\tau], \langle -1, \tau\rangle)$, where $\tau$ is the golden ratio, i.e. the positive root of $x^2 – x – 1 = 0$. There is a homomorphism to a number of fields, notably $\text{GF}(4)$ and $\text{GF}(5)$. In fact, it can be shown that a matroid has a golden ratio representation if and only if it is representable over both $\text{GF}(4)$ and $\text{GF}(5)$.

The partial field $\mathbb{P}_4$ is defined as follows. Let $\alpha$ be an indeterminate, let $G$ be the subgroup of the units of $\mathbb{Q}(\alpha)$ generated by $\{-1, \alpha, \alpha-1, \alpha + 1, \alpha – 2\}$, and let $R$ be the smallest subring of $\mathbb{Q}(\alpha)$ containing $G$. There is a partial field homomorphism to every field with at least four elements (obtained by mapping $\alpha$ to any element other than $0, 1, -1, 2$). Let me end with a conjecture about this partial field:

**Conjecture 15. **A matroid is representable over all fields with at least four elements if and only if it is representable over $\mathbb{P}_4$.

Note that the partial fields for the corresponding statements for two and three elements are known: they are, respectively, the *regular *and *near-regular* partial fields.

For a bigger catalog of partial fields, including a list of homomorphisms between them, I refer to the appendix of my PhD Thesis, [vZ09].

# References

[SW96] Charles Semple and Geoff Whittle. *Partial fields and matroid representation.* Adv. in Appl. Math., 17(2):184–208, 1996.

[PZ13] R.A. Pendavingh, S.H.M. van Zwam. *Skew partial fields, multilinear representations of matroids, and a matrix tree theorem. *Advances in Applied Mathematics, Vol. 50, Issue 1, pp. 201-227, 2013 (PDF, arXiv, doi).

[Whi95] Geoff Whittle. *A characterisation of the matroids representable over GF(3) and the rationals.* J. Combin. Theory Ser. B, 65(2):222–261, 1995.

[Whi97] Geoff Whittle. *On matroids representable over GF(3) and other fields.* Trans. Amer. Math. Soc., 349(2):579–603, 1997.

[vZ09] Stefan H. M. van Zwam. *Partial Fields in Matroid Theory*. PhD thesis, Technische Universiteit Eindhoven, 2009.