The Cone of Class Function Inequalities for the 4-By-4 Positive Semidefinite Matrices

Abstract

A function f from the symmetric group Sn into R is called a class function if f(σ) = f(τ) whenever τ is conjugate to σ. Let df be thegeneralized matrix function associated with f, mapping the n-by-n positive semidefinite Hermitian matricesto R. For example, if f(σ) = sgn(σ), then df(A) = det A. We consider the cone Kn of those f for which df(A) ⩾ 0 for all n-by-n positive semidefinite Hermitian matrices. For n = 4 we show that Kn is polyhedral, and explicitly find the extreme rays. Equivalently, f belongs to Kn if df(A) ⩾ 0 for a finite minimal set of ‘test matrices’. This solves a problem posed by Gordon James. As a corollary, we characterize all inequalities involving linear combinations of immanants on the positive semidefinite Hermitian matrices of order n = 4. We obtain similar results for 4-by-4 real positive semidefinite matrices. 1991 Mathematics Subject Classification: 15A15, 15A45, 15A48.