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

TL;DR

This paper introduces a higher rank Racah algebra as the symmetry algebra of the $ ext{Z}_2^n$ Laplace-Dunkl operator, linking it to superintegrable models on the sphere and constructing explicit Dunkl harmonic bases.

Contribution

It develops a higher rank generalization of the Racah algebra, constructs explicit Dunkl harmonic bases, and provides methods for connection coefficients and symmetry actions.

Findings

Established the higher rank Racah algebra as the symmetry algebra of the Laplace-Dunkl operator.

Constructed explicit bases of Dunkl harmonics using a Cauchy-Kovalevskaia theorem.

Presented methods for calculating connection coefficients and symmetry actions on these bases.

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.