Grassmann--Pl\"ucker functions for orthogonal matroids

TL;DR

This paper introduces a new cryptomorphic definition of orthogonal matroids using Grassmann--Pl"ucker functions, linking algebraic identities with geometric parameterizations.

Contribution

It provides a novel cryptomorphic characterization of orthogonal matroids via Grassmann--Pl"ucker functions, connecting minors of skew-symmetric matrices with geometric structures.

Findings

Orthogonal matroids can be characterized using Grassmann--Pl"ucker functions.

Each component of the orthogonal Grassmannian is parameterized by specific Pl"ucker coordinates.

Cayley's identities relate minors of skew-symmetric matrices to Pfaffians.

Abstract

We present a new cryptomorphic definition of orthogonal matroids with coefficients using Grassmann--Pl\"ucker 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 Pl\"ucker coordinates.