A nonpolyhedral cone of class function inequalities for positive semidefinite matrices

Abstract

A function f from the symmetric group Sn into R 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 R. For example, if f(σ)=sgn(σ), then df(A)=detA. Let Kn(Kn(R)) denote the closed convex cone of those f for which df(A)⩾0 for all n-by-n positive semidefinite Hermitian (real symmetric) matrices. For n=1,2,3,4 it is known that Kn and Kn(R) are polyhedral and there is a finite set of “test” matrices Tn(Tn(R)) such that f belongs to Kn(Kn(R)) if and only if df(A)⩾0 for each A in Tn(Tn(R)). We show here that K5 and K5(R) 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.