Grassmann--Plücker functions for orthogonal matroids
Changxin Ding, Donggyu Kim
TL;DR
The paper develops a new cryptomorphic framework for orthogonal matroids over tracts by introducing restricted Grassmann–Plücker functions of type $D$, establishing bijections with Wick $F$-matroids and orthogonal $F$-signatures. It specializes to the complex field via the restricted Plücker embedding of the orthogonal Grassmannian, clarifying the two connected components as parity choices and giving linear-relations descriptions for the restricted coordinates. It further shows that the support of a restricted GP function yields an underlying even antisymmetric matroid, yielding a natural bridge to orthogonal matroids; this leads to a robust nonrealizable-to-realizable correspondence and a complete set of cryptomorphisms, including a parallel weak theory. Additional results connect to enveloping matroids and Pfaffian positivity through Cayley identities, highlighting broader implications for positivity in the orthogonal setting and for the geometry of the orthogonal Grassmannian.
Abstract
We present a new cryptomorphic definition of orthogonal matroids with coefficients using Grassmann--Plücker functions. The equivalence is motivated by Cayley's identities expressing principal and almost-principal minors of a skew-symmetric matrix in terms of its Pfaffians. As a corollary of the new cryptomorphism, we deduce that each component of the orthogonal Grassmannian is parameterized by certain part of the Plücker coordinates.