Representability of orthogonal matroids over partial fields

Abstract

Let $r⩽n$ be nonnegative integers, and let $N=\left(\genfrac{}{}{0ex}{}{n}{r}\right)-1$. 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 $p=(p_J)\in {\mathbf{P}}^N(k)$ with support $M$ (meaning that $\text{Supp}(p):=\{J\in (\genfrac{}{}{0ex}{}{E}{r})|p_J\ne 0\}$ of $p$ is the set of bases of $M$) satisfying the Grassmann-Plücker equations; and (c) there is a point $p=(p_J)\in {\mathbf{P}}^N(k)$ with support $M$ satisfying just the 3-term Grassmann-Plücker 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.

13 pages