Zonal polynomial

In mathematics, a zonal polynomial is a multivariate symmetric homogeneous polynomial. The zonal polynomials form a basis of the space of symmetric polynomials. Zonal polynomials appear in special functions with matrix argument which on the other hand appear in matrixvariate distributions such as the Wishart distribution when integrating over compact Lie groups. The theory was started in multivariate statistics in the 1960s and 1970s in a series of papers by Alan Treleven James and his doctoral student Alan Graham Constantine.{{cite journal|first=Alan Treleven|last=James |title=Zonal Polynomials of the Real Positive Definite Symmetric Matrices|journal=Annals of Mathematics |volume=74 |date=1961 |issue=3 |doi=10.2307/1970291 |pages=456–469|jstor=1970291 }}{{cite journal|first=Alan Treleven|last=James |title=Distributions of Matrix Variates and Latents Roots Derived from Normal Samples|journal=Ann. Math. Statist.|volume=35 |date=1964|issue=2 |pages=475–501 |doi= 10.1214/aoms/1177703550|doi-access=free}}{{cite journal|first=Alan Graham|last=Constantine|title=Some Noncentral Distribution Problems in Multivariate Analysis|journal=Ann. Math. Statist.|volume=34 |date=1963 |issue=4 |pages=1270–1285 |doi=10.1214/aoms/1177703863|doi-access=free}}

They appear as zonal spherical functions of the Gelfand pairs

$(S\_\{2n\},H\_n)$ (here, $H\_n$ is the hyperoctahedral group) and $(Gl\_n(\backslash mathbb\{R\}),$

O_n), which means that they describe canonical basis of the double class

algebras $\backslash mathbb\{C\}[H\_n\; \backslash backslash\; S\_\{2n\}\; /\; H\_n]$ and $\backslash mathbb\{C\}[O\_d(\backslash mathbb\{R\})\backslash backslash$

M_d(\mathbb{R})/O_d(\mathbb{R})].

The zonal polynomials are the $\backslash alpha=2$ case of the C normalization of the Jack function.