A nonpolyhedral cone of class function inequalities for positive semidefinite matrices

A function f from the symmetric group S-n into R is called a class function if it is constant on each conjugacy class. Let df be the generalized matri x function associated with f, mapping the n-by-n Hermitian matrices to R. F or example, if f(sigma) = sgn(sigma), then d(f)(A) = det A. Let K-n(K-n (R) ) denote the closed convex cone of those f for which d(f) (A) greater than or equal to 0 for all n-by-n positive semidefinite Hermitian (real symmetri c) matrices. For n = 1, 2, 3, 4 it is known that K-n and K-n(R) are polyhed ral and there is a finite set of "test" matrices T-n(T-n(R)) such that f be longs to K-n(K-n(R)) if and only if d(f) (A) greater than or equal to 0 for each A in T-n(T-n(R)). We show here that K-5 and K-5(R) are not polyhedral . Thus, for n = 5 there is no finite set of "test" matrices sufficient to e stablish which generalized matrix functions are nonnegative on the positive semidefinite matrices. (C) 1999 Elsevier Science Inc. All rights reserved. AMS classification: 15A15; 15A45; 15A48.