Extreme rays for cones of Hermitian matrix functions that are non-negative on the positive semi-definite matrices (Q1300856)

Let \(K_n\) be the cone of complex-valued Hermitian functions \(\lambda\) on the symmetric group \(S_n\) such that the generalized matrix function \(d_\lambda\) is non-negative on the positive semi-definite matrices. It is shown that for \(n\geq 3\) the cone \(K_n\) is non-polyhedral -- an infinite set of extremal rays is constructed. This should be compared with a result of \textit{W. Barrett, H. T. Hall} and \textit{R. Loewy} [The cone of class function inequalities for the 4-by-4 positive semidefinite matrices. Proc. Lond. Math. Soc., III. Ser. 79, No.~1, 107-130 (1999; Zbl 1024.15008)] that if \(n\leq 4\) then the cone of class-functions in \(K_n\) is finitely generated.