THE DETERMINANTAL CONJECTURE AND HADAMARD TYPE INEQUALITIES

The first part of this paper presents an approach to a possible salvat ion of an idea advanced by papers of Bebiano, Merikoski, da Providenci a, and Virtanen for the solution of the Oliveira-Marcus determinantal conjecture. Let a(1),...,a(n) and b(1),...,b(n) be given complex numbe rs, and define the vertex points upsilon(sigma) = Pi(j = 1)(n)(a(j) b(sigma(j))) is an element of C for sigma a permutation. Let A and B b e normal matrices with eigenvalues a(1),...,a(n) and b(1),...,b(n) res pectively. The Oliveira-Marcus conjecture asks whether the complex num ber det( A + B) is necessarily in the convex hull of the n! vertex poi nts. The authors mentioned above had hoped that for every unitary matr ix U, the ((n)(k)) x ((n)(k)) nonnegative matrix with entries \det U[I , J]\(2) might be representable as convex combinations of the subset o f matrices stemming from permutation matrices. Unfortunately this turn s out not to be the case. The second part shows how inequalities of Ha damard type can be used to further this program at least in the specia l case n = 4. The Oliveira-Marcus conjecture is reduced in case n = 4 to a geometrical problem relating only to the disposition of the verte x points.