Representability of orthogonal matroids over partial fields
Matthew Baker, Tong Jin
TL;DR
The paper extends the Grassmannian/Plücker framework for matroid representability to orthogonal matroids (Lagrangian orthogonal matroids) by introducing Wick equations and Pfaffian representations. It proves that representability over partial fields for orthogonal matroids is equivalent to Pfaffian representations of skew-symmetric matrices up to a twist, mirroring the Plücker-based equivalences for ordinary matroids. The authors establish normal-form equivalences, connect these objects to orthogonal Grassmannians, and provide a Nelson-type asymptotic non-representability result: asymptotically 100% of orthogonal matroids are not representable over any field, hence over any partial field. Consequently, representability over partial fields collapses to representability over some field, yielding analogous scarcity in both ordinary and orthogonal settings.
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-Plücker equations; and (c) there is a point $p = (p_J) \in {\bf P}^N(k)$ 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.