New duality operator for complex circulant matrices and a conjecture of Ryser (Q276213)

Summary: 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 \(C_4\) on their eigenvalues, namely, the absolute value of the sum of all eigenvalues is bounded above by \(2\). We prove by a ``descent'' that uses our operator \(E\) that the only circulant Hadamard matrices of order \(n \ge 4\), that satisfy a condition \(C_n\) that generalizes the condition \(C_4\) and that consist of a list of \(n/4\) inequalities for the absolute value of some sums of eigenvalues of \(H\) are the known ones.