Racah Polynomials and Recoupling Schemes of $\mathfrak{su}(1,1)$

Abstract

The connection between the recoupling scheme of four copies of $\mathfrak{su}(1,1)$, the generic superintegrable system on the 3 sphere, and bivariate Racah polynomials is identified. The Racah polynomials are presented as connection coefficients between eigenfunctions separated in different spherical coordinate systems and equivalently as different irreducible decompositions of the tensor product representations. As a consequence of the model, an extension of the quadratic algebra $\mathrm{QR}(3)$ is given. It is shown that this algebra closes only with the inclusion of an additional shift operator, beyond the eigenvalue operators for the bivariate Racah polynomials, whose polynomial eigenfunctions are determined. The duality between the variables and the degrees, and hence the bispectrality of the polynomials, is interpreted in terms of expansion coefficients of the separated solutions.

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Cited by in corpus (9)

• A higher rank Racah algebra and the ${\mathbb{Z}}_2^n$ Laplace-Dunkl operator
• Representations of the rank two Racah algebra and orthogonal multivariate polynomials
• FRT presentation of classical Askey-Wilson algebras
• A higher rank extension of the Askey-Wilson Algebra
• Coupling coefficients of $s{u}_q(1,1)$ and multivariate $q$-Racah polynomials
• A bispectral q-hypergeometric basis for a class of quantum integrable models
• Construction of polynomial algebras from intermediate Casimir invariants of Lie algebras
• $\lambda$-Griffiths polynomials: Bispectrality and biorthogonality
• Matrix elements of $SO(3)$ in $s{l}_3$ representations as bispectral multivariate functions