A matroid or oriented matroid is dyadic if it has a rational representation with all nonzero subdeterminants in [ \2 k : 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.
1999 Academic Press
A rational matrix is totally dyadic if all of its nonzero subdeterminants are in D :=[\2 k : k # Z]. A matroid or oriented matroid is dyadic if it can be represented over Q by a totally dyadic matrix. It is easy to see that dyadic matroids are ternary, since elements of D map to nonzero elements of GF(3) when viewed modulo 3. Hence the matroids that underlie dyadic oriented-matroids are ternary. Our main result is the if '' part of the following theorem (the only if '' part being the simple observation mentioned above).
Theorem 1. Let M be an oriented matroid. Let M be the matroid underlying M. Then M is dyadic if and only if M is ternary.
This theorem resolves a conjecture of J. Lee (Conjecture III.5.5c of [6]) in the positive. We have the following two consequences of our main result.
Corollary 1. Let M be a ternary matroid. If M is orientable, then M is dyadic.