New Duality Operator for Complex Circulant Matrices and a Conjecture of Ryser

We associate to any given circulant complex matrix C another one E(C) such that E(E(C)) = C∗ the transpose conjugate of C. All circulant Hadamard matrices of order 4 satisfy a condition C4 on their eigenvalues, namely, the absolute value of the sum of all eigenvalues is bounded above by 4. We prove by a “descent” that uses our operator E that the only circulant Hadamard matrices of order n > 4, that satisfy a condition Cn that generalizes the condition C4 and that consist of a list of n/4 inequalities for the absolute value of some sums of four eigenvalues of H, are the known ones.