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In mathematics, a zonal polynomial is a multivariate symmetric homogeneous polynomial. The zonal polynomials form a basis of the space of symmetric polynomials.

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

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

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+In [[mathematics]], a '''zonal polynomial''' is a multivariate [[symmetric polynomial|symmetric]] [[homogeneous polynomial]]. The zonal polynomials form a [[basis (algebra)|basis]] of the space of symmetric polynomials.
+They appear as [[zonal spherical function]]s of the [[Gelfand pair]]s
+<math>(S_{2n},H_n)</math> (here, <math>H_n</math> is the hyperoctahedral group) and <math>(Gl_n(\mathbb{R}),
+O_n)</math>, which means that they describe canonical basis of the double class
+algebras <math>\mathbb{C}[H_n \backslash S_{2n} / H_n]</math> and  <math>\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>\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.
+{{algebra-stub}}
+[[Category:Homogeneous polynomials]]
+[[Category:Symmetric functions]]