Representability of orthogonal matroids over partial fields

Abstract

Let r≤n be nonnegative integers, and let N=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 )∈P N (k) with support M (meaning that Supp(p):={J∈E r|p J ≠0} of p is the set of bases of M) satisfying the Grassmann-Plücker equations; and (c) there is a point p=(p J )∈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.