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A discrete realization of the higher rank Racah algebra

Hendrik De Bie (UGent) and Wouter van de Vijver
(2020) CONSTRUCTIVE APPROXIMATION. 52(1). p.1-29
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Abstract
In previous work, a higher rank generalizationR(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 byR(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 ofR(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 ofR(n) as a discrete operator acting on the multivariate Racah polynomials.
Keywords
Analysis, General Mathematics, Computational Mathematics, Racah algebra, Racah polynomials, Superintegrable system, Coupling coefficients, Difference operators, ORTHOGONAL POLYNOMIALS, SUPERINTEGRABLE SYSTEM

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MLA
De Bie, Hendrik, and Wouter van de Vijver. “A Discrete Realization of the Higher Rank Racah Algebra.” CONSTRUCTIVE APPROXIMATION, vol. 52, no. 1, 2020, pp. 1–29, doi:10.1007/s00365-019-09475-0.
APA
De Bie, H., & van de Vijver, W. (2020). A discrete realization of the higher rank Racah algebra. CONSTRUCTIVE APPROXIMATION, 52(1), 1–29. https://doi.org/10.1007/s00365-019-09475-0
Chicago author-date
De Bie, Hendrik, and Wouter van de Vijver. 2020. “A Discrete Realization of the Higher Rank Racah Algebra.” CONSTRUCTIVE APPROXIMATION 52 (1): 1–29. https://doi.org/10.1007/s00365-019-09475-0.
Chicago author-date (all authors)
De Bie, Hendrik, and Wouter van de Vijver. 2020. “A Discrete Realization of the Higher Rank Racah Algebra.” CONSTRUCTIVE APPROXIMATION 52 (1): 1–29. doi:10.1007/s00365-019-09475-0.
Vancouver
1.
De Bie H, van de Vijver W. A discrete realization of the higher rank Racah algebra. CONSTRUCTIVE APPROXIMATION. 2020;52(1):1–29.
IEEE
[1]
H. De Bie and W. van de Vijver, “A discrete realization of the higher rank Racah algebra,” CONSTRUCTIVE APPROXIMATION, vol. 52, no. 1, pp. 1–29, 2020.
@article{8665674,
  abstract     = {{In previous work, a higher rank generalizationR(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 byR(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 ofR(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 ofR(n) as a discrete operator acting on the multivariate Racah polynomials.}},
  author       = {{De Bie, Hendrik and van de Vijver, Wouter}},
  issn         = {{0176-4276}},
  journal      = {{CONSTRUCTIVE APPROXIMATION}},
  keywords     = {{Analysis,General Mathematics,Computational Mathematics,Racah algebra,Racah polynomials,Superintegrable system,Coupling coefficients,Difference operators,ORTHOGONAL POLYNOMIALS,SUPERINTEGRABLE SYSTEM}},
  language     = {{eng}},
  number       = {{1}},
  pages        = {{1--29}},
  title        = {{A discrete realization of the higher rank Racah algebra}},
  url          = {{http://doi.org/10.1007/s00365-019-09475-0}},
  volume       = {{52}},
  year         = {{2020}},
}
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