Representability of orthogonal matroids over partial fields

Let $r \leqslant n$ be nonnegative integers, and let $N = \binom{n}{r} - 1$. For a matroid $M$ of rank $r$ on the finite set $E = [n]$ and a partial field $k$ in the sense of Semple--Whittle, it is known that the following are equivalent: (a) $M$ is representable over $k$; (b) there is a point $p = (p_J) \in {\bf P}^N(k)$ with support $M$ (meaning that $\text{Supp}(p) := \{J \in \binom{E}{r} \; \vert \; p_J \ne 0\}$ of $p$ is the set of bases of $M$) satisfying the Grassmann-Pl\"ucker equations; and (c) there is a point $p = (p_J) \in {\bf P}^N(k)$ with support $M$ satisfying just the 3-term Grassmann-Pl\"ucker equations. Moreover, by a theorem of P. Nelson, almost all matroids (meaning asymptotically 100%) are not representable over any partial field. We prove analogues of these facts for Lagrangian orthogonal matroids in the sense of Gelfand-Serganova, which are equivalent to even Delta-matroids in the sense of Bouchet.


Introduction
For simplicity, throughout this introduction k will denote a field, but all statements remain true when k is a partial field in the sense of Semple and Whittle [13], and the proofs will be written in that generality.
Let E be a finite set of size n, which for concreteness we will sometimes identify with the set [n] := {1, 2, . . ., n}.Let r be a nonnegative integer, and let N = n r − 1.Let E r denote the family of all r-subsets of E. We will be considering the projective space P N (k) with coordinates indexed by the r-element subsets of E.
Let A be an r × n matrix of rank r whose columns are indexed by E. Define ∆ : E r → k by ∆(J) = det A J , where A J is the r × r maximal square submatrix whose set of columns is J.We can extend the map ∆ to all subsets of E of size at most r by setting ∆(J) = 0 if |J| < r.For every S ∈ E r+1 , T ∈ E r−1 , and x ∈ S, we define sgn(x; S, T ) to be (−1) m , where m is the number of elements s ∈ S with s > x plus the number of elements t ∈ T with t > x.Basic properties of determinants imply that the point p = (p J ) J∈( E r ) ∈ P N (k) defined by p J := ∆(J) satisfies the following homogeneous quadratic equations, called the Plücker equations (cf.e.g.[8, §4.3]): (1) When S\T = {i < j < k}, we may assume without loss of generality that T \S = {ℓ} and ℓ > k.Then we obtain the 3-term Plücker equations, which are of particular importance: (2) X S\{i} X T ∪{i} − X S\{j} X T ∪{j} + X S\{k} X T ∪{k} = 0.
The following result is fundamental: Theorem 1.1.Let k be a field.The following are equivalent for a point p = (p J ) ∈ P N (k): (1) There exists an r × n matrix A of rank r with entries in k such that p J = det(A J ) for any J ∈ E r .
(2) The point p satisfies the Plücker equations.
(3) The support Supp(p) = {J ∈ E r | p J ̸ = 0} of p is the set of bases of a matroid of rank r on E, and p satisfies the 3-term Plücker equations.
The equivalence of ( 1) and ( 2) in the theorem is just the well-known classical fact that the Grassmannian variety of r-dimensional subspaces of a fixed n-dimensional vector space is defined (set-theoretically) by the Plücker equations.The equivalence of ( 2) and ( 3) is a folklore fact, but we are not aware of a published reference which furnishes a direct proof, so we give one in Section 2 below.
One of the interesting features of Theorem 1.1 is that it is a 'purely algebraic' fact about matrices over a field but its statement involves the combinatorial notion of a matroid.Recall that a matroid M on E is a nonempty collection B of subsets of E satisfying the following exchange axiom: A matroid whose bases are the support of some point p = (p J ) ∈ P N (k) satisfying the Plücker equations (or, equivalently, the 3-term Plücker equations) is called representable over k.
It is natural to ask whether a 'typical' matroid is representable over some field.The answer turns out to be no.This follows by combining the following two estimates: Theorem 1.2 (Knuth [7]).Denote by m n the number of isomorphism classes of matroids on ground set [n] = {1, 2, . . ., n}.Then (Here log is taken to base two.) Theorem 1.3 (Nelson [10]).For n ⩾ 12, the number r n of isomorphism classes of matroids on the ground set [n] which are representable over some field satisfies Combining these two estimates, we obtain: Theorem 1.4.Asymptotically 100% of all matroids are not representable over any field.
One can generalize the classical Grassmannian varieties by the Lagrangian orthogonal Grassmannians OG(n, 2n), which parameterise all n-dimensional maximal isotropic subspaces of a given 2n-dimensional vector space endowed with a symmetric, non-degenerate bilinear form.The combinatorial counterpart of this point of view is the notion of a Lagrangian orthogonal matroid, also known as an even Delta-matroid [4].For simplicity, we omit the adjective 'Lagrangian' and refer to such objects as orthogonal matroids.
Definition 1.5.Denote by X∆Y the symmetric difference of two sets X, Y .An orthogonal matroid on E is a nonempty collection B of subsets of E satisfying the following axiom: if B 1 , B 2 ∈ B, then for any x 1 ∈ B 1 ∆B 2 , there exists an element Like the usual Grassmannian, the Lagrangian orthogonal Grassmannian OG(n, 2n) is also a projective variety cut out by homogeneous quadratic polynomials, known in this case as the Wick equations [12,14] (see equations (4) below for a precise formulation).The simplest Wick equations have precisely four non-zero terms.
By analogy with Theorem 1.1, we will prove: Theorem 1.6.Let k be a field. (1)Let n ∈ N, set E = [n] and N = 2 n − 1, and consider the projective space P N (k) with coordinates indexed by the subsets of [n].
The following are equivalent for a point p = (p J ) ∈ P N (k).
(1) There exists a skew-symmetric matrix A over k with rows and columns indexed by E and a subset T ⊆ E such that p J = Pf(A J∆T ) for all J ⊆ E. (Here A J denotes the |J| × |J| square submatrix whose sets of row and column indices are both J, and Pf denotes the Pfaffian of a skew-symmetric matrix.)(2) The point p satisfies the Wick equations.
(3) The support of p is the set of bases of an orthogonal matroid on E, and p satisfies the 4-term Wick equations.
In particular, a point p ∈ P N (k) belongs to OG(n, 2n) if and only if there is a subset T ⊆ E such that Supp(p)∆T is the set of bases of an orthogonal matroid and p satisfies the 4-term Wick relations.
If M is an orthogonal matroid, we say that M is representable over k if there is a skew-symmetric matrix A over k with rows and columns indexed by E and a subset T ⊆ E such that p J = Pf(A J∆T ) for all J ⊆ E.
By analogy with Nelson's theorem, we estimate the number of representable orthogonal matroids and show: Theorem 1.7.Asymptotically 100% of orthogonal matroids are not representable over any field.

Representations of Matroids over Partial Fields
Partial fields are generalizations of fields which have proven to be very useful for studying representability of matroids.They were originally introduced by Semple and Whitte [13], but the definitions below are from [11]. (2)  Definition 2.1.A partial field P is a pair (G, R) consisting of a commutative ring R with 1 and a subgroup G of the group of units of R such that −1 belongs to G. We say p is an element of P and write p ∈ P if p ∈ G ∪ {0}. (1)In §4, we will generalize this fact, and the statement of Theorem 1.6, to partial fields k. (2) In [2], one finds a slightly different definition of partial fields from the one in [11] (there is an additional requirement that G generates R as a ring), but the difference is irrelevant for our purposes in this paper.
Example 2.2.A partial field with G = R\{0} is the same thing as a field.
Example 2.3.The partial field F ± 1 = ({1, −1}, Z) is called the regular partial field.Definition 2.4.Let P = (G, R) be a partial field.A strong P -matroid of rank r on E = [n] is a projective solution p = (p J ) ∈ P N (P ) to the Plücker equations (1), i.e. p J ∈ P for all J ∈ E r , not all P J are zero, and p satisfies (1) viewed as equations over R. A weak P -matroid of rank r on E = [n] is a projective solution p = (p J ) to the 3-term Plücker equations (2) such that Supp(p) := {J | p J ̸ = 0} is the set of bases of a matroid of rank r on E.
Remark 2.5.Let P be a partial field.If M is a strong or weak P -matroid, corresponding to a Plücker vector p ∈ P N (P ), then Supp(p) := {J | p J ̸ = 0} is the set of bases of a matroid M on E, called the underlying matroid of M .We say that a matroid M is P -representable (or representable over P ) if it is the support of a strong (or, equivalently, by the following theorem, weak) P -matroid.
The following is a generalization of Theorem 1.1 to partial fields: Then the following are equivalent for a nonzero point p = (p J ) ∈ P N (P ): (1) There exists a matrix A with entries in P such that p J = det(A J ) for all J ∈ [n] r .
(2) p satisfies the Plücker equations (1), i.e., p is the Plücker vector of a strong P -matroid.(3) p is the Plücker vector of a weak P -matroid.
Proof.It follows from standard properties of determinants for matrices over commutative rings that (1) implies (2), and (2) implies ( 3) is true by definition.It remains to show that (3) implies (1).The idea is that given p : [n]   r → P whose support is the set of bases of a matroid M , we will explicitly construct an r × n matrix A over P such that det(A J ) = p(J) for all r-element subsets J ⊆ [n].
Without loss of generality, by relabeling the elements of E and rescaling the projective vector p if necessary, we may assume that [r] = {1, 2, . . ., r} is a basis of M and that p([r]) = 1.We define where a ij = (−1) r+i p([r]\{i} ∪ {j}).We claim that det(A J ) = p(J) for all J ∈ [n]  r .The proof is by induction on v , and elementary properties of determinants give det(A J ) = (−1) i+r a ij = p(J).
We wish to show that det(A J ) = p(J).
Case 1. Suppose J is a basis of M .Then by the basis exchange property, there exists i ∈ [r]\J such that B ′ := J\{a} ∪ {i} is a basis.By the basis exchange property again, there exists j ∈ [r]\B ′ such that B ′′ := B ′ \{b}∪{j} is also a basis.Without loss of generality, we may assume that i < j, and then applying the 3-term Grassmann-Plücker relations to S = J\{b} ∪ {i, j} and T = J\{a} (so that S\T = {i < j < a}), we obtain Moreover, since S\{a} = B ′′ is a basis, p(S\{a}) = det(A S\{a} ) ̸ = 0.This gives p(T ∪ {a}) = det(A T ∪{a} ) as desired.
Case 2. Suppose J is not a basis of M , i.e., p(J) = 0. Note that if there exist distinct i, j ∈ [r]\J such that J\{a, b} ∪ {i, j} is a basis, then the proof from Case 1 still works.Therefore, we may assume that no such i and j exist.Then for every i ∈ [r]\J, J ′ i := J\{a} ∪ {i} is not a basis and det(A Together with (3), this forces det(A J ) = 0 = p J .□ We now explain briefly how to see that Theorem 1.4 (Nelson's theorem) implies that asymptotically 100% of all matroids are not representable over any partial field.Definition 2.7.
On the other hand, we also have: Lemma 2.9.If P = (G, R) is a partial field, there exists a homomorphism P → k for some field k.
Proof.Take a maximal ideal m ⊆ R and consider the field k := R/m.Then the natural quotient homomorphism R ↠ k induces a homomorphism of partial fields P → k. □ We deduce: Corollary 2.10.If a matroid is representable over a partial field P , then it is representable over a field k.
Combining this fact with Theorem 1.4, we obtain: Corollary 2.11.Asymptotically 100% of all matroids are not representable over any partial field.

Representations of Orthogonal Matroids
In this section, we provide some background on orthogonal matroids and their representations.
Recall from Section 1 that an orthogonal matroid on E is a nonempty collection B of subsets of E satisfying the symmetric exchange axiom.As with their matroid counterparts, this axiom can be replaced with an a priori stricter strong symmetric exchange axiom (see, for example, [3, Theorem 4.2.4]): Example 3.2.Every matroid is also an orthogonal matroid.In fact, matroids are by definition just orthogonal matroids whose bases all have the same cardinality.
From the definition, one sees easily that any two bases of an orthogonal matroid have the same parity.
The determinant of a matrix A admits a refinement for skew-symmetric matrices called the Pfaffian.The Pfaffian Pf(A) can be defined recursively as follows.
By convention, we define the Pfaffian of the empty matrix to be 1.Now let A be an n × n skew-symmetric matrix over a ring R, where n ⩾ 1.If n is odd, we set Pf A basic fact about the Pfaffians is that (Pf(A)) 2 = det(A) for every skew-symmetric matrix A [5].
Proposition 3.4 (Wenzel [14,Prop. 2.3 ]).Let A be an n × n skew-symmetric matrix over a ring R. Let N = 2 n − 1.The point (p J ) J⊆E ∈ P N (R) defined by p J = Pf(A J ) satisfies the homogeneous quadratic polynomial equations called the Wick equations, (3) We are especially interested in the shortest possible Wick equations, where Here, and from now on, Ja means J ∪ {a} in order to simplify the notation.If P is a partial field, we may consider the projective solutions in P N (P ) to the different kinds of Wick equations. (3)These identities are known to physicists as Wick's theorem [9].We follow [12] rather than [14] in our terminology.
Algebraic Combinatorics, Vol. 6 #5 (2023) Definition 3.5.A strong orthogonal matroid over P is a projective solution p = (p J ) to the Wick equations (4).A weak orthogonal matroid over P is a projective solution p = (p J ) to 4-term Wick equations whose support Supp(p) = {J | p J ̸ = 0} is the set of bases of an orthogonal matroid on E. Remark 3.6.One can generalize the definition of weak and strong orthogonal matroids over P from partial fields to tracts in the sense of [1] and obtain cryptomorphic descriptions of these objects along the lines of loc.cit., see [6].
Proposition 3.7 (Wenzel [14, Theorem 2.2]).Let P = (G, R) be a partial field.Given a strong orthogonal matroid over P with p ∅ = 1, there exists an n×n skew-symmetric matrix A over R such that Pf(A J ) = p J .Proposition 3.8.Let P be a partial field.The support of every strong orthogonal matroid over P is the set of bases of an orthogonal matroid.
Proof.If p ∅ = 1, then Supp(p) gives an orthogonal matroid by Theorem 3.3 of [14].Otherwise, let p ∈ P N (P ) be the Wick vector of a strong orthogonal matroid over P (i.e., a point of the Lagrangian orthogonal Grassmannian OG(n, 2n) over P ), and choose T ̸ = ∅ such that p T = 1.Consider the point q ∈ P N (P ) whose coordinates are defined by q J = p J∆T .We claim that q also satisfies the Wick equations.In fact, for any Since ∅ ∈ Supp(q), this gives an orthogonal matroid M with set of bases Supp(p)∆T .Therefore, Supp(p) is the set of bases for the twist M ∆T .□ Definition 3.9.Let M be an orthogonal matroid, and let P be a partial field.Then M is P -representable if there exists a skew-symmetric matrix A = (a ij ) i,j∈E with entries in P and a subset T ⊆ E such that Example 3.10.The two orthogonal matroids in Example 3.3 are both R-representable.
To see this, let Using the same notation from Example 3.3, we find that B = B ′ ∆{3} = {∅, {1, 2}, {1, 4}, {2, 4}}.Remark 3.11.In general, one can choose T = ∅ in the representation if and only if ∅ is a basis.In this case, we say that the orthogonal matroid is normal.

Orthogonal Matroids and Orthogonal Grassmannians
Let P be a partial field and let N = 2 n − 1.Our goal for this section is to connect the notions of strong orthogonal matroids, weak orthogonal matroids, and representable matroids over P .
We begin with the following lemma on normal orthogonal matroids.Proof.We apply the symmetric exchange axiom to the bases J and ∅.Pick some a ∈ J. Then there exists a ̸ = b ∈ J = J∆∅ such that J ′ = J∆{a, b} is a basis, and Using this lemma, we now prove the desired result for normal orthogonal matroids.Given a point p = (p J ) ∈ P N (P ) with support equal to the orthogonal matroid M and satisfying the 4-term Wick equations, let A = (a ij ) be the skew-symmetric matrix defined by Since p satisfies the 4-term Wick equations, we have But p J = Pf(A J ) when |J| = 2. Therefore, by the recursive definition of Pfaffians, p J = Pf(J) when |J| = 4. Now suppose p J = Pf(A J ) for bases J with |J| = 0, 2, 4, . . ., 2r − 2. Let J = {i 1 , . . ., i 2r } be a basis of M with 1 ⩽ i 1 < • • • < i 2r ⩽ n.By Lemma 4.1, there exists another basis J ′ ⊆ J with |J ′ | = 2r − 4. Without loss of generality J\J ′ = {i 1 , i 2 , i 3 , i 4 }.
Finally, suppose J = {i 1 , . . ., i 2r } ⊆ [n] is not a basis for M .If there exists {a, b, c, d} ⊆ J such that J ′ = J\{a, b, c, d} is a basis, then the same proof would apply, giving Pf(A J ) = p J = 0. Otherwise, we have p We now turn to the general case. Therefore, By Theorem 5. Proof.Let A = (a ij ) be a skew-symmetric matrix over P 1 and T a subset of E such that the pair (A, T ) represents M .Consider the new matrix B = φ(A) := (φ(a ij )).If Pf(A J ) = 0 then Pf(B J ) = φ(0) = 0 ∈ P 2 .If Pf(A J ) ̸ = 0, then Pf(B J ) = φ(Pf(A J )) ̸ = 0.This shows that the pair (B, T ) represents M .□ Combined with Lemma 2.9, this yields: Corollary 5.5.If an orthogonal matroid is representable over a partial field P , then it is representable over some field k.
We conclude: Corollary 5.6.Asymptotically 100% of all orthogonal matroids are not representable over any partial field.
then for any x ∈ B 1 \B 2 , there exists an element y ∈ B 2 \B 1 such that (B 1 \{x}) ∪ {y} belongs to B. This turns out to be equivalent to the a priori stricter strong exchange axiom: If B 1 , B 2 ∈ B, then for any x ∈ B 1 \B 2 , there exists an element y ∈ B 2 \B 1 such that (B 1 \{x}) ∪ {y} and (B 2 \{y}) ∪ {x} both belong to B. The set E is called the ground set of M , and the members of B are called the bases.All bases of a matroid M have the same cardinality, called the rank of M .
1, {f 1 , . . ., f N } has at most r = 2 n 3 realizable sets, so there are at most 2 n 3 normal orthogonal matroids on [n] which are representable over some field.□ Corollary 5.3.Asymptotically 100% of all orthogonal matroids are not representable over any field.Proof.Let k be a field and let be a k-representable orthogonal matroid.Then for any T ∈ B, M ∆T is a normal representable orthogonal matroid.This shows the number of representable orthogonal matroid on [n] is at most 2 n • 2 n 3 .The result now follows from Theorem 1.2.
□Lemma 5.4.Let φ : P 1 → P 2 be a homomorphism of partial fields.If an orthogonal matroid M is P 1 -representable, then M is also P 2 -representable.