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 2 n , H n ) (here, H n is the hyperoctahedral group) and ( G l n ( R ) , O n ) , which means that they describe canonical basis of the double class algebras C [ H n ∖ S 2 n / H n ] and C [ O d ( R ) ∖ M d ( R ) / O d ( R ) ] .

They are applied in multivariate statistics.

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