A Characterization of the Matroids Representable over GF(3) and the Rationals

It follows from a fundamental (1958) result of Tutte that a binary matroid is representable over the rationals if and only if it can be represented by a totally unimodular matrix, that is, by a matrix over the rationals with the property that all subdeterminants belong to {0, 1, −1}. For an arbitrary field F, it is of interest to ask for a matrix characterisation of those matroids representable over F and the rationals. In this paper this question is answered when F is GF(3). It is shown that a ternary matroid is representable over the rationals if and only if it can be represented over the rationals by a matrix A with the property that all subdeterminants of A belong to the set {0, ±2i:i an integer}. While ternary matroids are uniquely representable over GF(3), this is not generally the ease for representations of ternary matroids over other fields. A characterisation is given of the class of ternary matroids that are uniquely representable over the rationals.