Zonal polynomials and hypergeometric functions of quaternion matrix argument

TL;DR

This paper introduces zonal polynomials for quaternion matrices, derives key formulas for hypergeometric functions, and applies these to determine eigenvalue distributions of quaternion Wishart matrices.

Contribution

It defines quaternion zonal polynomials, derives fundamental formulas, and applies them to eigenvalue distribution problems in quaternion Wishart matrices.

Findings

Derived distributions of largest and smallest eigenvalues of quaternion Wishart matrices

Established formulas for zonal polynomials of quaternion matrices

Extended hypergeometric functions to quaternion matrix arguments

Abstract

We define zonal polynomials of quaternion matrix argument and deduce some important formulae of zonal polynomials and hypergeometric functions of quaternion matrix argument. As an application, we give the distributions of the largest and smallest eigenvalues of a quaternion central Wishart matrix $W\sim\mathbb{Q}W(n,\Sigma)$, respectively.