Cones in the group algebra related to Schur's determinantal inequality (Q1107596)

If c is a complex valued function on the symmetric group \(S_ n\) then \(d_ c(A)\) is defined by \[ d_ c(A)=\sum_{\sigma \in S_ n}c(\sigma)\prod^{n}_{t=1}a_{t\sigma (t)}, \] when \(A=(a_{ij})\) is an \(n\times n\) matrix. If C is the cone of functions c with \(d_ c(A)\geq 0\) for all positive semidefinite A then it is shown that \(d_ c(A)\geq c(e)\det (A)\) for all \(c\in C\) and all positive semidefinite A. [That generalizes classical results of Hadamard and Schur and a recent result of \textit{R. B. Bapat} and \textit{V. S. Sunder}, Linear Algebra Appl. 76, 153-163 (1986; Zbl 0602.15008)]. The result is used to study the cone C in more detail.