A discrete realization of the higher rank Racah algebra

TL;DR

This paper provides a concrete discrete realization of the higher rank Racah algebra by connecting it to multivariate Racah polynomials through Dunkl-harmonics and explicit operator actions.

Contribution

It introduces a new explicit realization of the higher rank Racah algebra using Dunkl models and multivariate Racah polynomials, extending previous abstract definitions.

Findings

Connection coefficients are multivariate Racah polynomials.

Action of the Racah algebra corresponds to Racah operators on these polynomials.

Generators of the algebra act as discrete operators on multivariate Racah polynomials.

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 to 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 subalgebra 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.