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September 10, 2025Open Access

Orthogonal matroids over tracts

TJTong JinGeorgia Institute of Technology DKDonggyu KimPohang University of Science and Technology

Key Points

Orthogonal matroids over tracts demonstrate significant representation properties, revealing their structural uniqueness.
The research establishes that an orthogonal matroid is regular if it can be represented over both ${ extmath{F}}_2$ and ${ extmath{F}}_3$.
Using definitions involving Wick functions and vector sets, the paper proves several representation theorems related to orthogonal matroids.
Findings suggest important connections between representation over partial fields and established characteristics of orthogonal matroids.

Abstract

Abstract We generalize Baker–Bowler’s theory of matroids over tracts to orthogonal matroids, define orthogonal matroids with coefficients in tracts in terms of Wick functions, orthogonal signatures, circuit sets and orthogonal vector sets, and establish basic properties on functoriality, duality and minors. Our cryptomorphic definitions of orthogonal matroids over tracts provide proofs of several representation theorems for orthogonal matroids. In particular, we give a new proof that an orthogonal matroid is regular if and only if it is representable over F₂ and F₃, which was originally shown by Geelen 16, and we prove that an orthogonal matroid is representable over the sixth-root-of-unity partial field if and only if it is representable over F₃ and F₄.

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