An extremal property of the permanent and the determinant (Q1081661)

Let \(A=[a_{ij}]\) be an \(n\times n\) complex matrix. Let \(\tilde A\) be the n!\(\times n!\) matrix whose rows and columns are indexed with the elements of \(S_ n\) (the symmetric group of degree n), the (\(\sigma\),\(\tau)\) entry of \(\tilde A\) being \(\prod^{n}_{i=1}a_{\tau (i),\sigma (i)}\). Assume A is Hermitian positive semidefinite. The authors give an elementary proof of a result of Schur stating that det A is the smallest eigenvalue of \(\tilde A\) and also prove a somewhat stronger result. They consider the conjecture that per A is the largest eigenvalue of \(\tilde A\) and prove it for \(n\leq 3\). Other conjectures are presented.