Abstract :
Given a complex valued function λ with domain sn, the symmetric group on {1,2,…,n), we define the matrix function dλ(·) in the usual way, and restrict dλ(·) to script capital hn, the n × n positive semi-definite Hermitian matrices. If λset membership, variantthe field of complex numbersSn,. the algebra of functions from Sn to the field of complex numbers. then λ is said to be Hermitian if λ(σ−1 = λ(σ) for each σ set membership, variant Sn. By Kn we mean the cone of Hermitian elements λ set membership, variantthe field of complex numbersSn such that dλ{A) greater-or-equal, slanted0 for each A set membership, variantscript capital hn, and by Keln, also a cone, we mean the intersection of Kn with the set of class functions on Sn. Recently Barrett, Hall, and Loewy showed that if nless-than-or-equals, slant4. then Keln is finitely generated, and that there is a finite set the geometric imaginary line.Un subset of ∛n of matrices, called test matrices, such that a class function λ: Sn → the field of complex numbers is in Keln if and only if dλ{A) greater-or-equal, slanted 0 for each A set membership, variant the geometric imaginary line.Un. We demonstrate that if n greater-or-equal, slanted 3. then the larger cone Kn is not finitely generated by presenting an infinite set of extreme rays. Consequently, there is no finite set of test matrices for Kn when n greater-or-equal, slanted 3. In addition we present a scheme that is effective in identifying extreme rays in both Kn and Kefn 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 Kn, and Kefn.
