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An explicit formula for zonal polynomials

Haoming Wang

Abstract

The derivation of zonal polynomials involves evaluating the integral \[ \exp\left( - \frac{1}{2} \operatorname{tr} D_β Q D_{l} Q \right) \] with respect to orthogonal matrices \(Q\), where \(D_β\) 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.

An explicit formula for zonal polynomials

Abstract

The derivation of zonal polynomials involves evaluating the integral with respect to orthogonal matrices , where and 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 . 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.

Paper Structure

This paper contains 13 sections, 4 theorems, 74 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Characters of symmetric groups
  3. Polar coordinates for orthogonal matrices
  4. Derivations of zonal polynomials
  5. Orthonormal systems in the space of symmetric orthogonal matrices
  6. Spherical harmonics
  7. Calculating spherical harmonics by solving linear systems
  8. Zonal polynomials
  9. Relations to partial differential equations
  10. An explicit formula for zonal polynomials
  11. Applications
  12. Distribution of latent roots of the non-central Wishart distribution
  13. Zonal polynomials up to $f=6$ degrees

Key Result

Theorem 2.1

Let $h_1(x), \ldots, h_n(x)$ be real analytic functions. Then,

Figures (1)

  • Figure 1: Orthogonal decomposition in terms of rotation and reflection. The blue one is the original vector and the red one is the target vector under rotation and reflection.

Theorems & Definitions (4)

  • Theorem 2.1
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 5.1