A nonpolyhedral cone of class function inequalities for positive semidefinite matrices Original Research Article

A function f from the symmetric group Sn into image is called a class function if it is constant on each conjugacy class. Let df be the generalized matrix function associated with f, mapping the n-by-n Hermitian matrices to image . For example, if f(σ)=sgn(σ), then df(A)=detA. Let image denote the closed convex cone of those f for which df(A)greater-or-equal, slanted0 for all n-by-n positive semidefinite Hermitian (real symmetric) matrices. For n=1,2,3,4 it is known that Kn and image are polyhedral and there is a finite set of “test” matrices image such that f belongs to image if and only if df(A)greater-or-equal, slanted0 for each A in image . We show here that K5 and image are not polyhedral. Thus, for n=5 there is no finite set of “test” matrices sufficient to establish which generalized matrix functions are nonnegative on the positive semidefinite matrices.