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

Symmetry Integrability and Geometry-Methods and Applications

article

Sarah Post¹ 

[1] Department of Mathematics, University of Hawai`i at Mānoa

关键词: orthogonal polynomials;    Lie algebras;    representation theory;   

DOI  :  10.3842/SIGMA.2015.057

来源: National Academy of Science of Ukraine

【 摘 要 】

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 ${\rm QR}(3)$ 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.

【 授权许可】

Unknown