A higher rank Racah algebra and the $\mathbb{Z}_2^{n}$ Laplace-Dunkl operator

Abstract

A higher rank generalization of the (rank one) Racah algebra is obtained as the symmetry algebra of the Laplace-Dunkl operator associated to the $\mathbb{Z}_2^n$ root system. This algebra is also the invariance algebra of the generic superintegrable model on the $n$-sphere. Bases of Dunkl harmonics are constructed explicitly using a Cauchy-Kovalevskaia theorem. These bases consist of joint eigenfunctions of maximal Abelian subalgebras of the higher rank Racah algebra. A method to obtain expressions for both the connection coefficients between these bases and the action of the symmetries on these bases is presented.