A counterexample to a question of Merikoski and Virtanen on the compounds of unitary matrices (Q1189632)

Let \(U\) be a unitary \(n\times n\) matrix. The conjecture of \textit{J. K. Merikovki} and \textit{A. Virtanen} [ibid. 121, 345-352 (1989; Zbl 0692.15003)] mentioned in the title claims that \(| U^{(m)}|^ 2\) is a convex combination of \(| P_ \sigma^{(m)}|\) for each \(m=1,2,\dots,n\), where \((m)\) denotes the \(m\)-th compound, \(P_ \sigma\) is the permutation matrix corresponding to a permutation \(\sigma\) of \(\{1,2,\dots,n\}\), and the functions \(|\cdot|\) and \(|\cdot|^ 2\) are to be understood elementwise. The conjecture is shown to be false for \(n=4\). This, however, does not disprove the conjecture of \textit{M. Marcus} [Indiana Univ. Math. J. 22, 1137-1149 (1973; Zbl 0243.15025)] and \textit{G. N. de Oliveira} [Research problem: Normal matrices, Linear Multilinear Algebra 12, 153-154 (1982)] on the location of \(\text{det}(A+B)\) with normal matrices \(A\) and \(B\), which was in turn motivated by the precise result of \textit{M. Fiedler} [Proc. Am. Math. Soc. 30, 27-31 (1971; Zbl 0277.15010)] in the case where the above matrices \(A\) and \(B\) are Hermitian.