An explicit formula for zonal polynomials

TL;DR

This paper derives an explicit formula for zonal polynomials by evaluating integrals involving orthogonal matrices, providing a systematic polynomial expansion and clarifying their fundamental properties.

Contribution

It introduces a new explicit formula for zonal polynomials based on integral evaluations, enhancing understanding of their structure and properties.

Findings

Explicit polynomial expansion for zonal polynomials

Systematic derivation of coefficients from integral structure

Confirmation of uniqueness up to normalization

Abstract

The derivation of zonal polynomials involves evaluating the integral \[ \exp\left( - \frac{1}{2} \operatorname{tr} D_{\beta} Q D_{l} Q \right) \] with respect to orthogonal matrices \(Q\), where \(D_{\beta}\) and \(D_{l}\) are diagonal matrices. The integral is expressed through a polynomial expansion in terms of the traces of these matrices, leading to the identification of zonal polynomials as symmetric, homogeneous functions of the variables \(l_1, l_2, \ldots, l_n\). The coefficients of these polynomials are derived systematically from the structure of the integrals, revealing relationships between them and illustrating the significance of symmetry in their formulation. Furthermore, properties such as the uniqueness up to normalization are established, reinforcing the foundational role of zonal polynomials in statistical and mathematical applications involving orthogonal matrices.