Representability of orthogonal matroids over partial fields

TL;DR

This paper extends the understanding of when orthogonal matroids can be represented over partial fields, showing parallels to classical matroid representability and highlighting that most matroids are non-representable.

Contribution

It establishes analogues of classical matroid representability criteria for Lagrangian orthogonal matroids, connecting them to Grassmann-Plücker equations and partial field representations.

Findings

Almost all matroids are not representable over any partial field.

Representability over partial fields is characterized by solutions to Grassmann-Plücker equations.

Analogous criteria are proven for Lagrangian orthogonal matroids, linking them to even Delta-matroids.

Abstract

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.