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.

They appear as zonal spherical functions of the Gelfand pairs (S_{2n},H_n) (here, H_n is the hyperoctahedral group) and (Gl_n(\mathbb{R}),
O_n), which means that they describe canonical basis of the double class algebras \mathbb{C}[H_n \backslash S_{2n} / H_n] and \mathbb{C}[O_d(\mathbb{R})\backslash
M_d(\mathbb{R})/O_d(\mathbb{R})].

They are applied in multivariate statistics.

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

References