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Canonical forms of oriented matroids

Christopher Eur, Thomas Lam

TL;DR

This work extends the notion of canonical forms from realizable positive geometries to oriented matroids by constructing ${\overline \Omega}(\mathcal{M},\chi)$ in the reduced Orlik--Solomon algebra and its tope version ${\overline \Omega}(P,\chi)$, satisfying a residue-based recursion that mirrors positive geometry axioms. It then builds a canonical-form basis for the Orlik--Solomon algebra and the Aomoto cohomology via truncations and general extensions, recovering classical chamber bases in the realizable case (notably Yoshinaga’s basis) up to a scalar. In the generic Aomoto setting, the bounded-tope canonical forms provide a basis for $H^{r-1}({\overline{{\operatorname{OS}}}}^\bullet,\omega)$, with dimension given by the matroid beta invariant and (in the realizable case) connections to twisted cohomology. The paper also offers concrete rank-2 and rank-3 examples illustrating the construction and its compatibility with known positive geometry structures, forming a foundational tool for matroid amplitudes and related combinatorial-geometric frameworks.

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.

Canonical forms of oriented matroids

TL;DR

This work extends the notion of canonical forms from realizable positive geometries to oriented matroids by constructing in the reduced Orlik--Solomon algebra and its tope version , satisfying a residue-based recursion that mirrors positive geometry axioms. It then builds a canonical-form basis for the Orlik--Solomon algebra and the Aomoto cohomology via truncations and general extensions, recovering classical chamber bases in the realizable case (notably Yoshinaga’s basis) up to a scalar. In the generic Aomoto setting, the bounded-tope canonical forms provide a basis for , with dimension given by the matroid beta invariant and (in the realizable case) connections to twisted cohomology. The paper also offers concrete rank-2 and rank-3 examples illustrating the construction and its compatibility with known positive geometry structures, forming a foundational tool for matroid amplitudes and related combinatorial-geometric frameworks.

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.

Paper Structure

This paper contains 10 sections, 22 theorems, 49 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Canonical forms of oriented matroids and topes
  3. Realizable case
  4. Definition of positive geometry
  5. Hyperplane arrangements
  6. Basis for the Orlik--Solomon algebra
  7. Basis for Aomoto cohomology
  8. Examples
  9. Rank two
  10. Rank three

Key Result

Proposition 2.1

Dimca We have that ${\overline{{\operatorname{OS}}}}_R^\bullet(\mathrm{M}) = \partial ({\operatorname{OS}}_R^\bullet(\mathrm{M}))$, and the map $\partial: {\operatorname{OS}}_R^r(\mathrm{M}) \longrightarrow {\overline{{\operatorname{OS}}}}_R^{r-1}(\mathrm{M})$ is an isomorphism of $R$-modules.

Figures (1)

  • Figure 1: Left: the affine diagram of the arrangement of points in $\mathbb P(V^\vee)$. Right: the corresponding hyperplane arrangement in $\mathbb R^2 \cong \mathbb P(V)\setminus \mathbb P H_q$.

Theorems & Definitions (47)

  • Proposition 2.1
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • Lemma 2.6
  • proof
  • proof : Proof of Theorem \ref{['thm:main']}
  • ...and 37 more