Canonical forms of oriented matroids

TL;DR

This paper introduces a canonical form for any tope of an oriented matroid within the Orlik--Solomon algebra, extending the concept of positive geometries to combinatorial structures and enabling new bases for algebraic and cohomological theories.

Contribution

It constructs a canonical form for topes of oriented matroids and develops bases for the Orlik--Solomon algebra and Aomoto cohomology, advancing matroid amplitude theory.

Findings

Canonical forms for topes are constructed within the Orlik--Solomon algebra.

New bases for the Orlik--Solomon algebra and Aomoto cohomology are established.

The work provides foundational tools for the theory of matroid amplitudes.

Abstract

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.