Zonal polynomial

Last updated March 01, 2019

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

Mathematics field of study concerning quantity, patterns and change

Mathematics includes the study of such topics as quantity, structure, space, and change.

In mathematics, a symmetric polynomial is a polynomial $P (X 1, X 2, \dots, X n)$ in $n$ variables, such that if any of the variables are interchanged, one obtains the same polynomial. Formally, $P$ is a symmetric polynomial if for any permutation $σ$ of the subscripts $1, 2, ..., n$ one has $P (X σ, X σ, \dots, X σ) = P (X 1, X 2, \dots, X n)$ .

In mathematics, a homogeneous polynomial is a polynomial whose nonzero terms all have the same degree. For example, $is a homogeneous polynomial of degree 5, in two variables; the sum of the exponents in each term is always 5. The polynomial is not homogeneous, because the sum of exponents does not match from term to term. A polynomial is homogeneous if and only if it defines a homogeneous function. An algebraic form, or simply form, is a function defined by a homogeneous polynomial. A binary form is a form in two variables. A form is also a function defined on a vector space, which may be expressed as a homogeneous function of the coordinates over any basis.$

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 $\mathbb {C} [O_{d}(\mathbb {R} )\backslash M_{d}(\mathbb {R} )/O_{d}(\mathbb {R} )]$ .

In mathematics, a zonal spherical function or often just spherical function is a function on a locally compact group G with compact subgroup K that arises as the matrix coefficient of a K-invariant vector in an irreducible representation of G. The key examples are the matrix coefficients of the spherical principal series, the irreducible representations appearing in the decomposition of the unitary representation of G on L²(G/K). In this case the commutant of G is generated by the algebra of biinvariant functions on G with respect to K acting by right convolution. It is commutative if in addition G/K is a symmetric space, for example when G is a connected semisimple Lie group with finite centre and K is a maximal compact subgroup. The matrix coefficients of the spherical principal series describe precisely the spectrum of the corresponding C* algebra generated by the biinvariant functions of compact support, often called a Hecke algebra. The spectrum of the commutative Banach *-algebra of biinvariant L¹ functions is larger; when G is a semisimple Lie group with maximal compact subgroup K, additional characters come from matrix coefficients of the complementary series, obtained by analytic continuation of the spherical principal series.

In mathematics, the expression Gelfand pair is a pair (G,K) consisting of a group G and a subgroup K that satisfies a certain property on restricted representations. The theory of Gelfand pairs is closely related to the topic of spherical functions in the classical theory of special functions, and to the theory of Riemannian symmetric spaces in differential geometry. Broadly speaking, the theory exists to abstract from these theories their content in terms of harmonic analysis and representation theory.

They are applied in multivariate statistics.

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

In mathematics, the Jack function is a generalization of the Jack polynomial, introduced by Henry Jack. The Jack polynomial is a homogeneous, symmetric polynomial which generalizes the Schur and zonal polynomials, and is in turn generalized by the Heckman–Opdam polynomials and Macdonald polynomials.

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References

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

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