A characterisation 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 \({\mathbf F}\), it is of interest to ask for a matrix characterisation of those matroids representable over \({\mathbf F}\) and the rationals. In this paper this question is answered when \({\mathbf F}\) 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 \(\{0, \pm 2^i : i\) an integer\}. While ternary matroids are uniquely representable over GF(3), this is not generally the case 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.
scientific article; zbMATH DE number 839123