Grassmann--Plücker functions for orthogonal matroids

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.

💡 Research Summary

This paper introduces a new cryptomorphic description of orthogonal matroids (also called even Δ‑matroids) by means of restricted Grassmann–Plücker functions, and proves that this description is equivalent to the previously known Wick‑function definition. The authors work in the general setting of tracts, a class of algebraic objects introduced by Baker and Bowler that generalize fields, sign hyperfields, tropical hyperfields, etc., by keeping a multiplicative group structure while weakening addition to a “zero‑test” via a null set.

In the classical theory, a matroid over a tract F can be encoded by a Grassmann–Plücker function φ: (\binom{E}{r}) → F satisfying (GP1) non‑vanishing and (GP2) the usual Grassmann–Plücker exchange relations. Orthogonal matroids, which model isotropic subspaces of a 2n‑dimensional space equipped with a symmetric bilinear form, have been described by Wick functions ψ: Tₙ → F (where Tₙ is the set of transversals of (