Extreme rays and faces for the cone of class functions non-negative on the set of Gram matrices Original Research Article

If f is a image-valued function with domain image, the symmetric group on {1,2,…,m}, then the matrix function [f](·), or df(·), is defined by [f](A)=∑σf(σ)∏t=1mat,σ(t) for all m×m complex matrices A=[aij]. We consider the cone Kmcℓ whose elements are the Hermitian class functions image such that [f](A)greater-or-equal, slanted0 for each image, where image denotes the set of all m×m positive semi-definite Hermitian matrices. The extreme rays of Kcℓm are fundamental to an understanding of the linear inequalities that result by restricting the various [f](·) to the sets image. In particular, the resolution of the permanent dominance conjecture for immanants and certain related conjectures such as the conjectures of Lieb and Soules will likely involve identification and analysis of these rays. Barrett, Hall, and Loewy gave a complete list of the extreme rays of Kcℓm when mless-than-or-equals, slant4, and have shown that K5cℓ is not polyhedral. Given positive integers n and p such that nless-than-or-equals, slantp and n+p=m, we let Kcℓn,p denote the subcone of Kmcℓ consisting of all fset membership, variantKcℓm such that f is expressible as a linear combination of the irreducible characters of image associated with partitions of the form (2i,1m−2i) where 0less-than-or-equals, slantiless-than-or-equals, slantn. We show that Kcℓn,p is an extreme polyhedral subcone, or face, of Kcℓm, and give explicit formulas for each of its n+1 extreme rays. Thus, Kmcℓ has non-trivial polyhedral faces for all m.