Extreme rays and faces for the cone of class functions non-negative on the set of Gram matrices (Q1886521)

Let \(f\) be a complex valued function with domain \(\mathbb{S}_m,\) the symmetric group on \(\{1,2,\ldots ,m\}.\) The matrix function \([f](\cdot )\) associated with \(f\) is defined by \([f](A)=\sum_{\sigma }f(\sigma )\prod_{t=1}^{m}a_{t,\sigma (t)}\) for all \(m\times m\) matrices \(A=[a_{ij}].\) Consider the cone \({\mathbf K}_m ^{ cl}\) whose elements are the Hermitian class functions \(f:\mathbb{S}_m\rightarrow \mathbb{C}\) such that \([f](A)\geq 0\) for each \(A\in {\mathcal H}_m,\) where \({\mathcal H}_m\) denotes the set of all \(m\times m\) positive semi-definite Hermitian matrices. \textit{W. Barrett, H. T. Hall} and \textit{R. Loewy} [Proc. London Math. Soc. 79, 107-130 (1999; Zbl 1024.15008), Linear Algebra Appl. 302-303, 535-553 (1999; Zbl 0959.15025)] proved that \({\mathbf K}_m^{ cl}\) is polyhedral and gave a complete list of its extreme rays when \(m\leq 4\) and have shown that \({\mathbf K}_5^{ cl}\) is not polyhedral. Let \(n\) and \(p\) be positive integers such that \(n\leq p\) and let \(m=n+p.\) Let \({\mathbf K}_{n,p}^{ cl}\) denote the subcone of \({\mathbf K}_m^{ cl}\) consisting of all \(f\in {\mathbf K}_m^{ cl}\) such that \(f\) is expressible as a linear combination of the irreducible characters of \(\mathbb{S}_m\) associated with partitions of the form \((2^i,1^{m-2i})\) where \(0\leq i\leq n.\) The author proves that \({\mathbf K}_{n,p}^{ cl}\) is an extreme polyhedral subcone or face of \({\mathbf K}_m^{ cl}\)and gives explicit formulas for each of its \(n+1\) extreme rays.