Methods and Applications of Analysis

Multidimensional Hahn polynomials, intertwining functions on the symmetric group and Clebsch-Gordon coefficients

Fabio Scarabotti

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Abstract

We generalize a construction of Dunkl, obtaining a wide class intertwining functions on the symmetric group Sn and a related family of multidimensional Hahn polynomials. Following a suggestion of Vilenkin and Klimyk, we develop a tree-method approach for those intertwining functions. Moreover, using our theory of $S_n$-intertwining functions and James version of the Schur- Weyl duality, we give a proof of the relation between Hahn polynomials and $SU(2)$ Clebsch-Gordan coefficients, previously obtained by Koornwinder and by Nikiforov, Smorodinskiĭ and Suslov in the $SU(2)$-setting. Such relation is also extended to the multidimensional case.

Article information

Source
Methods Appl. Anal., Volume 14, Number 4 (2007), 355-386.

Dates
First available in Project Euclid: 4 November 2008

Permanent link to this document
https://projecteuclid.org/euclid.maa/1225813982

Mathematical Reviews number (MathSciNet)
MR2467106

Zentralblatt MATH identifier
1177.33030

Subjects
Primary: 33C80: Connections with groups and algebras, and related topics
Secondary: 20C30: Representations of finite symmetric groups 33C45: Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) [See also 42C05 for general orthogonal polynomials and functions] 33C50: Orthogonal polynomials and functions in several variables expressible in terms of special functions in one variable 81R05: Finite-dimensional groups and algebras motivated by physics and their representations [See also 20C35, 22E70]

Keywords
Hahn polynomials intertwining functions tree method symmetric group special unitary group Clebsch-Gordan coefficients $3nj$-coefficients

Citation

Scarabotti, Fabio. Multidimensional Hahn polynomials, intertwining functions on the symmetric group and Clebsch-Gordon coefficients. Methods Appl. Anal. 14 (2007), no. 4, 355--386. https://projecteuclid.org/euclid.maa/1225813982


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