Abstract :
The first part of this paper presents an approach to a possible salvation of an idea advanced by papers of Bebiano, Merikoski, da Providência, and Virtanen for the solution of the Oliveira-Marcus determinantal conjecture. Let a1,…,an and b1,…,bn be given complex numbers, and define the vertex points v(θ) = ∏nj = 1(aj + bθ(j)) ε C for θ a permutation. Let A and B be normal matrices with eigenvalues a1,…,an and b1,…,bn respectively. The Oliveira-Marcus conjecture asks whether the complex number det(A + B) is necessarily in the convex hull of the n! vertex points. The authors mentioned above had hoped that for every unitary matrix U, the image nonegative matrix with entries det U[I, J]2 might be representable as convex combinations of the subset of matrices stemming from permutation matrices. Unfortunately this turns out not to be the case. The second part shows how inequalities of Hadamard type can be used to further this program at least in the special case n = 4. The Oliveira-Marcus conjecture is reduced in case n = 4 to a geometrical problem relating only to the disposition of the vertex points.
