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

Abstract

Given a complex valued function λ with domain s n , the symmetric group on {1,2,…, n ), we define the matrix function d λ (·) in the usual way, and restrict d λ (·) to ℋ n , the n × n positive semi-definite Hermitian matrices. If λ∈ℂS n ,. the algebra of functions from S n to ℂ. then λ is said to be Hermitian if λ(σ −1 = λ(σ) for each σ ∈ S n . By K n we mean the cone of Hermitian elements λ ∈ℂS n such that d λ {A) ⩾0 for each A ∈ℋ n , and by K el n , also a cone, we mean the intersection of K n with the set of class functions on S n . Recently Barrett, Hall, and Loewy showed that if n ⩽4. then K el n is finitely generated, and that there is a finite set ℐ.U n ⊂ ∛ n of matrices, called test matrices, such that a class function λ: S n → ℂ is in K el n if and only if d λ { A ) ⩾ 0 for each A ∈ ℐ.U n . We demonstrate that if n ⩾ 3. then the larger cone K n is not finitely generated by presenting an infinite set of extreme rays. Consequently, there is no finite set of test matrices for K n when n ⩾ 3. In addition we present a scheme that is effective in identifying extreme rays in both K n and K ef n and use it in conjunction with various sets of test matrices to identify additional extreme rays in each of these cones. In particular, certain of the rays associated with the Fischer inequality are proved to be extreme in K n , and K ef n .