The cone of positive generalized matrix functions (Q1209395)

For \(c:S_ n\to\mathbb{C}\), a complex valued function on the symmetric group \(S_ n\) and \(A=(a_{ij})\in\mathbb{C}^{n\times n}\), the author considers the determinant like polynomial \(d(c,A)=\sum_ \sigma c(\sigma)\prod^ n_{t=1}a_{t\sigma(t)}\). The cone \({\mathcal C}_ n\) of such functions which are positive on Hermitian matrices gets special attention. A complete description of \({\mathcal C}_ 4\) is given. For the general case some families of extreme rays are investigated.