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

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