Stiefel tropical linear spaces

The tropical Stiefel map associates to a tropical matrix A   its tropical Plücker vector of maximal minors, and thus a tropical linear space L(A)L(A). We call the L(A)L(A)s obtained in this way Stiefel tropical linear spaces  . We prove that they are dual to certain matroid subdivisions of polytopes of transversal matroids, and we relate their combinatorics to a canonically associated tropical hyperplane arrangement. We also explore a broad connection with the secondary fan of the Newton polytope of the product of all maximal minors of a matrix. In addition, we investigate the natural parametrization of L(A)L(A) arising from the tropical linear map defined by A.