The polytope of even doubly stochastic matrices

The polytope Q, of the convex combinations of the permutation matrices of order n is well known (Birkhoff’s theorem) to be the polytope of doubly stochastic matrices of order n. In particular it is easy to decide whether a matrix of order n belongs to Q,. . check to see that the entries are nonnegative and that all row and columns sums equal 1. Now the permutations z of { 1, 2, . . . . n} are in one-to-one correspondence with the permutation matrices P, of order n and we make speak of an even permutation matrix. Mirsky [4] defined a doubly stochastic matrix to be even provided it is a convex combination of even permutation matrices and considered the problem (proposed by A. J. Hoffman) of deciding when a doubly stochastic matrix is even. Let Sz; denote the polytope of even doubly