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• W2029843984 abstract "If f is a C-valued function with domain Sm, 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 f:Sm→C such that [f](A)⩾0 for each A∈Hm, where Hm 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 Hm. 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 m⩽4, and have shown that K5cℓ is not polyhedral. Given positive integers n and p such that n⩽p and n+p=m, we let Kcℓn,p denote the subcone of Kmcℓ consisting of all f∈Kcℓm such that f is expressible as a linear combination of the irreducible characters of Sm associated with partitions of the form (2i,1m−2i) where 0⩽i⩽n. 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." @default.
• W2029843984 created "2016-06-24" @default.
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• W2029843984 date "2004-10-01" @default.
• W2029843984 modified "2023-09-27" @default.
• W2029843984 title "Extreme rays and faces for the cone of class functions non-negative on the set of Gram matrices" @default.
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• W2029843984 doi "https://doi.org/10.1016/j.laa.2004.04.027" @default.
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