Convex Polyhedra of Doubly Stochastic Matrices. I. Applications of the Permanent Function

The permanent function is used to determine geometrical properties of the set 52, of all II x it nonnegative doubly stochastic matrices. If ,F is a face of Q, , then F corresponds to an n x n (0, I)-matrix A, where the permanent of A is the number of vertices of 3. I f A is fully indecomposable, then the dimension of 9 equals u(A) 2n + 1, where u(A) is the number of I’s in A. The only twodimensional faces of s2, are triangles and rectangles. For n > 6, G’s has four types of three-dimensional faces. The facets of the faces of Sz, are characterized. Faces of 52, which are simplices are determined. If .F is a face of Q, which is twoneighborly but not a simplex, then S has dimension 4 and six vertices. All kdimensional faces with k + 2 vertices are determined. The maximum number of vertices of a k-dimensional face is 2”. All k-dimensional faces with at least 2L-1 + 1 vertices are determined.