Abstract
In previous work, a higher rank generalization R(n) of the Racah algebra was defined abstractly. The special case of rank one encodes the bispectrality of the univariate Racah polynomials and is known to admit an explicit realization in terms of the operators associated with these polynomials. Starting from the Dunkl model for which we have an action by R(n) on the Dunkl-harmonics, we show that connection coefficients between bases of Dunkl-harmonics diagonalizing certain Abelian subalgebras are multivariate Racah polynomials. By lifting the action of R(n) to the connection coefficients, we identify the action of the Abelian subalgebras with the action of the Racah operators defined by J. S. Geronimo and P. Iliev. Making appropriate changes of basis, one can identify each generator of R(n) as a discrete operator acting on the multivariate Racah polynomials.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
De Bie, H., De Clercq, H. , van de Vijver, W.: The higher rank q-deformed Bannai–Ito and Askey–Wilson algebra (2018). arXiv:1805.06642
De Bie, H., Genest, V.X., Vinet, L., van de Vijver, W.: A higher rank Racah algebra and the \(({\mathbb{Z}}_2)^n\) Laplace–Dunkl operator. J. Phys. A: Math. Theor. 51, 025203 (2018)
Dunkl, C.F.: Differential-difference operators associated to reflection groups. Trans. Am. Math. Soc. 311, 167–183 (1989)
Dunkl, C.F., Xu, Y.: Orthogonal Polynomials of Several Variables, second edn. Cambridge University Press, Cambridge (2014)
Gao, S., Wang, Y., Hou, B.: The classification of Leonard triples of Racah type. Linear Algebra Appl. 439, 1834–1861 (2013)
Genest, V.X., Vinet, L., Zhedanov, A.: The equitable racah algebra from three \(\mathfrak{su}(1,1)\) algebras. J. Phys. A 47, 025203 (2014)
Genest, V.X., Vinet, L., Zhedanov, A.: Superintegrability in two dimensions and the Racah–Wilson algebra. Lett. Math. Phys. 104, 931–952 (2014)
Genest, V.X., Vinet, L., Zhedanov, A.: The Racah algebra and superintegrable models. J. Phys. Conf. Ser. 512, 012011 (2014)
Geronimo, J.S., Iliev, P.: Bispectrality of multivariable Racah–Wilson polynomials. Constr. Approx. 31, 417–457 (2010)
Granovskii, Y.A., Zhedanov, A.S.: Nature of the symmetry group of the \(6j\)-symbol. Sov. Phys. JETP 67, 1982–1985 (1988)
Iliev, P.: The generic quantum superintegrable system on the sphere and Racah operators. Lett. Math. Phys. 107(11), 2029–2045 (2017)
Iliev, P.: Symmetry algebra for the generic superintegrable system on the sphere. J. High Energy Phys. 44(2), front matter+22 pp (2018)
Iliev, P., Xu, Y.: Connection coefficients for classical orthogonal polynomials of several variables. Adv. Math. 310, 290–326 (2017)
Iliev, P., Xu, Y: Hahn polynomials on polyhedra and quantum integrability (2017). arXiv:1707.03843
Kalnins, E., Miller, W., Post, S.: Wilson polynomials and the generic superintegrable system on the 2-sphere. J. Phys. A 40, 11525 (2007)
Kalnins, E., Miller, W., Post, S.: Two-variable Wilson polynomials and the generic superintegrable system on the 3-sphere. SIGMA Symmetry Integr. Geom. Methods Appl. 7, 51 (2011)
Kalnins, E., Miller, W., Post, S.: Contractions of 2D 2nd order quantum superintegrable systems and the Askey scheme for hypergeometric orthogonal polynomials. SIGMA Symmetry Integr. Geom. Methods Appl. 9, 57 (2013)
Koekoek, R., Lesky, P.A., Swarttouw, R.F.: Hypergeometric Orthogonal Polynomials and Their \(q\)-Analogues. Springer, Berlin (2010)
Lévy-Leblond, J.-M., Lévy-Nahas, M.: Symmetrical coupling of three angular momenta. J. Math. Phys. 6, 1372–1380 (1965)
Post, S.: Racah polynomials and recoupling schemes of \({\mathfrak{su}}(1,1)\). SIGMA Symmetry Integr. Geom. Methods Appl. 11, 057 (2015)
Tratnik, M.V.: Some multivariable orthogonal polynomials of the Askey tableau-discrete families. J. Math. Phys. 32, 2337–2342 (1991)
Terwilliger, P.: The equitable presentation for the quantum group \(U_q(\mathfrak{g})\) associated with a symmetrizable Kac–Moody algebra \(\mathfrak{g}\). J. Algebra 298, 302–319 (2006)
Acknowledgements
We thank Luc Vinet for fruitful discussions and helpful comments. This work was supported by the Research Foundation Flanders (FWO) under Grant EOS 30889451. WVDV is grateful to the Fonds Professor Frans Wuytack for supporting his research.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Tom H. Koornwinder.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
De Bie, H., van de Vijver, W. A Discrete Realization of the Higher Rank Racah Algebra. Constr Approx 52, 1–29 (2020). https://doi.org/10.1007/s00365-019-09475-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00365-019-09475-0