A Characterization of the Orientations of Ternary Matroids

Abstract

A matroid or oriented matroid is dyadic if it has a rational representation with all nonzero subdeterminants in {±2k: k∈Z}. Our main theorem is that an oriented matroid is dyadic if and only if the underlying matroid is ternary. A consequence of our theorem is the recent result of G. Whittle that a rational matroid is dyadic if and only if it is ternary. Along the way, we establish that each whirl has three inequivalent orientations. Furthermore, except for the rank-3 whirl, no pair of these are isomorphic.