THE GEOMETRY OF PRODUCTS OF MINORS

We consider the Newton polytope Sigma(m, n) of the product of all mine rs of an m x n matrix of indeterminates. Using the fact that this poly tope is the secondary polytope of the product Delta(m-1) x Delta(n-1) of simplices, and thus has faces corresponding to coherent polyhedral subdivisions of Delta(m-1) x Delta(n-1), we study facets of Sigma(m, n ), which correspond to the coarsest, nontrivial such subdivisions. We make use of the relation between secondary and fiber polytopes, which in this case gives a representation of Sigma(m, n) as the Minkowski av erage of all m x n transportation polytopes.