Zonal polynomials and hypergeometric functions of quaternion matrix argument

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.

💡 Research Summary

The paper develops a comprehensive theory of zonal polynomials and hypergeometric functions when the argument is a quaternion matrix. After a brief review of the classical real‑ and complex‑valued cases, the authors introduce the necessary background on quaternion algebra, the space of quaternion‑Hermitian (self‑dual) matrices, and the symplectic group (Sp(p)) that plays the role of the orthogonal or unitary groups in the non‑commutative setting.

In Section 3 the authors define quaternion zonal polynomials (C_{\kappa}(X)) indexed by a partition (\kappa). They prove that these polynomials are invariant under the action of (Sp(p)) and can be expressed as a constant (c_{\kappa}) multiplied by the symmetric monomial (m_{\kappa}(\lambda)) in the eigenvalues (\lambda_1,\dots,\lambda_p) of (X). A generating function is established:
\