Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 11 (2015), 057, 17 pages      arXiv:1504.03705      https://doi.org/10.3842/SIGMA.2015.057
Contribution to the Special Issue on Exact Solvability and Symmetry Avatars in honour of Luc Vinet

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

Sarah Post
Department of Mathematics, University of Hawai`i at Mānoa, Honolulu, HI, 96822, USA

Received April 16, 2015, in final form July 14, 2015; Published online July 23, 2015

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 ${\rm QR}(3)$ 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.

Key words: orthogonal polynomials; Lie algebras; representation theory.

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References

  1. Capel J.J., Kress J.M., Invariant classification of second-order conformally flat superintegrable systems, J. Phys. A: Math. Theor. 47 (2014), 495202, 33 pages, arXiv:1406.3136.
  2. Capel J.J., Kress J.M., Post S., Invariant classification and limits of maximally superintegrable systems in 3D, SIGMA 11 (2015), 038, 17 pages, arXiv:1501.06601.
  3. Gao S., Wang Y., Hou B., The classification of Leonard triples of Racah type, Linear Algebra Appl. 439 (2013), 1834-1861.
  4. Genest V.X., Vinet L., The generic superintegrable system on the 3-sphere and the $9j$ symbols of ${\mathfrak{su}}(1,1)$, SIGMA 10 (2014), 108, 28 pages, arXiv:1404.0876.
  5. Genest V.X., Vinet L., The multivariate Hahn polynomials and the singular oscillator, J. Phys. A: Math. Theor. 47 (2014), 455201, 39 pages, arXiv:1406.6719.
  6. Genest V.X., Vinet L., Zhedanov A., Superintegrability in two dimensions and the Racah-Wilson algebra, Lett. Math. Phys. 104 (2014), 931-952, arXiv:1307.5539.
  7. Geronimo J.S., Iliev P., Bispectrality of multivariable Racah-Wilson polynomials, Constr. Approx. 31 (2010), 417-457, arXiv:0705.1469.
  8. Goodman R., Wallach N.R., Symmetry, representations, and invariants, Graduate Texts in Mathematics, Vol. 255, Springer, Dordrecht, 2009.
  9. Granovskiǐ Ya.I., Lutzenko I.M., Zhedanov A.S., Mutual integrability, quadratic algebras, and dynamical symmetry, Ann. Physics 217 (1992), 1-20.
  10. Humphreys J.E., Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, Vol. 9, Springer-Verlag, New York - Berlin, 1972.
  11. Iliev P., Bispectral commuting difference operators for multivariable Askey-Wilson polynomials, Trans. Amer. Math. Soc. 363 (2011), 1577-1598, arXiv:0801.4939.
  12. Kalnins E.G., Miller Jr. W., Quadratic algebra contractions and second-order superintegrable systems, Anal. Appl. (Singap.) 12 (2014), 583-612, arXiv:1401.0830.
  13. Kalnins E.G., Miller Jr. W., Post S., Two-variable Wilson polynomials and the generic superintegrable system on the 3-sphere, SIGMA 7 (2011), 051, 26 pages, arXiv:1010.3032.
  14. Kalnins E.G., Miller Jr. W., Post S., Contractions of 2D 2nd order quantum superintegrable systems and the Askey scheme for hypergeometric orthogonal polynomials, SIGMA 9 (2013), 057, 28 pages, arXiv:1212.4766.
  15. Koornwinder T.H., The relationship between Zhedanov's algebra ${\rm AW}(3)$ and the double affine Hecke algebra in the rank one case, SIGMA 3 (2007), 063, 15 pages, math.QA/0612730.
  16. Koornwinder T.H., Zhedanov's algebra $\rm AW(3)$ and the double affine Hecke algebra in the rank one case. II. The spherical subalgebra, SIGMA 4 (2008), 052, 17 pages, arXiv:0711.2320.
  17. Miller Jr. W., Li Q., Wilson polynomials/functions and intertwining operators for the generic quantum superintegrable system on the 2-sphere, J. Phys. Conf. Ser. 597 (2015), 012059, 11 pages, arXiv:1411.2112.
  18. Miller Jr. W., Post S., Winternitz P., Classical and quantum superintegrability with applications, J. Phys. A: Math. Theor. 46 (2013), 423001, 97 pages, arXiv:1309.2694.
  19. Terwilliger P., The universal Askey-Wilson algebra and the equitable presentation of $U_q({\mathfrak{sl}}_2)$, SIGMA 7 (2011), 099, 26 pages, arXiv:1107.3544.
  20. Tratnik M.V., Some multivariable orthogonal polynomials of the Askey tableau-discrete families, J. Math. Phys. 32 (1991), 2337-2342.
  21. Zhedanov A., ''Hidden symmetry'' of Askey-Wilson polynomials, Theoret. and Math. Phys. 89 (1991), 1146-1157.

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