Permanental mates of doubly stochastic matrices (Q1208277)

Let \(A\) and \(B\) be each \(n\times n\) and doubly stochastic. If \(\text{per}[rA+(1-r)B]=\text{per }A\) for all \(0\leq r\leq 1\), each of \(A\) and \(B\) is called a permanental mate of the other. Any such pair \(A\), \(B\) is called a permanental pair. The set of all permanental mates of \(A\) is denoted by \(M(A)\). It is shown that there exists a permanental pair of \(n\times n\) doubly stochastic matrices \(A,B,A\neq B\), such that \(A\) and \(B\) do not both minimize the permanent on any face of the set of \(n\times n\) doubly stochastic matrices for \(n\geq 3\). A conjecture of \textit{S. G. Hwang} [ibid. 140, 89-100 (1990; Zbl 0712.15017)] states that if \(A\) is \(n\times n\) doubly stochastic with \(n\geq 3\) and if \(M(A)\) is a convex set, then \(\dim M(A)\leq (n^ 2- 3n+2)/2\). This is shown to be false for \(n=3\). It is also shown that there is essentially a unique two-dimensional convex \(M(A)\) in the \(3\times 3\) doubly stochastic matrices.