Foundations for a Theory of 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 S 1 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 Delucchi [7].We further show that complex matroids cannot have vector axioms analogous to those for oriented matroids.

Matroids over partial hyperstructures

“…The proof of the following result diverges somewhat from the treatment of the analogous assertion in [AD12]. It remains to prove (DP3) (resp.…”

“…Relation to the work of Anderson and Delucchi. While the proofs of our main theorems are somewhat long and technical, in principle a great deal of the hard work has already been done in [AD12], so on a number of occasions we merely point out that a certain proof from [AD12] goes through mutatis mutandis in the general setting of matroids over tracts. (By way of contrast, the proofs in the standard works on oriented and valuated matroids tend to rely on special properties of the sign and tropical hyperfields which do not readily generalize.)…”

“…Proofs of the main theorems are deferred to Section 4. There are two brief Appendices at the end of the paper: in Appendix A we collect some errata from [AD12], and in Appendix B we present a simplified point of view on fuzzy rings written by Oliver Lorscheid.…”

“…This is equivalent to the axiom system for phased circuits given in [AD12] in the case of phased matroids (i.e., when F " P). Also, the F -circuit Z in (C3) 1 is unique.…”

“…Grassmann-Plücker functions. We now describe a cryptomorphic characterization of weak and strong matroids over a tract F in terms of Grassmann-Plücker functions (called "chirotopes" in the theory of oriented matroids and "phirotopes" in [AD12]). In addition to being interesting in its own right, this description will be crucial for establishing a duality theory for matroids over F .…”