An explicit formula for zonal polynomials

2024/10/17 by Haoming Wang, Wang, Haoming
Mathematics · #05E05 #15B10 #32A50 #Combinatorics (math.CO) #FOS: Mathematics #Mathematical functions and polynomials #Mathematics and Applications #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2410.13558

openalex publication_date 2024/10/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The derivation of zonal polynomials involves evaluating the integral exp( - (1)/(2) tr D_β Q D_l Q ) 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 \(l₁, l₂, …, l_n\). 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.

Related

• A note about zonal polynomials
• Orthogonal polynomials through the invariant theory of binary forms
• Topological expansion of the coefficients of zonal polynomials in genus one
• Jacobi polynomials on the simplex of Hermitian matrices and zonal spherical functions on partial flag manifolds
• Alternative Orthogonal Polynomials