A Discrete Realization of the Higher Rank Racah Algebra

  • Hendrik De Bie
  • Wouter van de VijverEmail author


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.


Racah algebra Racah polynomials Superintegrable system Coupling coefficients Difference operators 

Mathematics Subject Classification

33C50 33C80 47B39 81R10 81R12 



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.


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Authors and Affiliations

  1. 1.Department of Electronics and Information Systems Faculty of Engineering and ArchitectureGhent UniversityGhentBelgium

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