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 [math]\displaystyle{ (S_{2n},H_n) }[/math] (here, [math]\displaystyle{ H_n }[/math] is the hyperoctahedral group) and [math]\displaystyle{ (Gl_n(\mathbb{R}), O_n) }[/math], which means that they describe canonical basis of the double class algebras [math]\displaystyle{ \mathbb{C}[H_n \backslash S_{2n} / H_n] }[/math] and [math]\displaystyle{ \mathbb{C}[O_d(\mathbb{R})\backslash M_d(\mathbb{R})/O_d(\mathbb{R})] }[/math].

They are applied in multivariate statistics.

The zonal polynomials are the [math]\displaystyle{ \alpha=2 }[/math] case of the C normalization of the Jack function.

References

• Robb Muirhead, Aspects of Multivariate Statistical Theory, John Wiley & Sons, Inc., New York, 1984.