Foundations for a theory of complex matroids

TL;DR

This paper develops a combinatorial theory of complex matroids that generalizes oriented matroids to complex space, capturing linear dependency, orthogonality, and determinants with multiple axiomatizations and a canonical circle action.

Contribution

It introduces complex matroids with equivalent axiomatizations, linking them to properties of complex linear algebra and extending the oriented matroid framework.

Findings

Multiple equivalent axiomatizations of complex matroids

Complex matroids capture properties of complex linear dependency and orthogonality

Existence of a canonical circle action on complex matroids

Abstract

We explore a combinatorial theory of linear dependency in complex space, "complex matroids", with foundations analogous to those for oriented matroids. We give multiple equivalent axiomatizations of complex matroids, showing that this theory captures properties of linear dependency, orthogonality, and determinants over C in much the same way that oriented matroids capture the same properties over R. In addition, our complex matroids come with a canonical circle action analogous to the action of C* on a complex vector space. Our phirotopes (analogues of determinants) are the same as those studied previously by Below, Krummeck, and Richter-Gebert and by Delucchi. We further show that complex matroids cannot have vector axioms analogous to those for oriented matroids.