Extreme Points of the Vandermonde Determinant and Wishart Ensemble on Symmetric Cones

In this paper Muhumuza, Asaph KeikaraLundengård, KarlMalyarenko, AnatoliySilvestrov, SergeiMango, John MageroKakuba, Godwinwe demonstrate the extreme points of the Wishart joint eigenvalue probability distributions in higher dimension based on the boundary points of the symmetric cones in Jordan algebras. The extreme points of the Vandermonde determinant are defined to be a set of boundary points of the symmetric cones that occur in both the discrete and continuous part of the Gindikin set. The symmetric cones form a basis for the construction of the degenerate and non-degenerate Wishart ensembles in Herm(m,C)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{{Herm}}(m,\mathbb {C})$$\end{document}, Herm(m,H)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{{Herm}}(m,\mathbb {H})$$\end{document}, Herm(3,O)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{{Herm}}(3,\mathbb {O})$$\end{document} denotes respectively the Jordan algebra of all Hermitian matrices of size m×m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m\times m$$\end{document} with complex entries, the skew field H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {H}$$\end{document} of quaternions, and the algebra O\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {O}$$\end{document} of octonions.