Canonical forms of oriented matroids

Positive geometries are semialgebraic sets equipped with a canonical differential form whose residues mirror the boundary structure of the geometry. Every full‐dimensional projective polytope is a positive geometry. Motivated by the canonical forms of polytopes, we construct a canonical form for any tope of an oriented matroid inside the Orlik–Solomon algebra of the underlying matroid. Using these canonical forms, we construct bases for the Orlik–Solomon algebra of a matroid and for the Aomoto cohomology. These bases of canonical forms are a foundational input in the theory of matroid amplitudes introduced by the second author.