Scalar-linear solvability of matroidal networks associated with representable matroids

We study matroidal networks introduced by Dougherty et al., who showed that if a network is scalar-linearly solvable over some finite field, then the network is a matroidal network associated with a representable matroid over a finite field. In this paper, we prove the converse. It follows that a network is scalar-linearly solvable if and only if the network is a matroidal network associated with a representable matroid over a finite field and that determining scalar-linear solvability of a network is equivalent to finding a representable matroid over a finite field and a valid network-matroid mapping. As a consequence, we obtain a correspondence between scalar-linearly solvable networks and representable matroids over finite fields. We note that this result, combined with the construction method due to Dougherty et al., can generate potentially new scalar-linearly solvable networks.