Symmetry algebra for the generic superintegrable system on the sphere

TL;DR

This paper provides a detailed algebraic analysis of the symmetries of the quantum superintegrable system on the sphere, revealing the structure of its irreducible polynomial representations and their connection to multivariable orthogonal polynomials.

Contribution

It characterizes the irreducible representations of the symmetry algebra and describes their action via multivariable Racah operators, linking algebraic and polynomial structures.

Findings

Maximal abelian subalgebras have unique eigenfunctions of Jacobi polynomials.

Representations on fixed-degree polynomial spaces are irreducible.

Symmetry actions are described using multivariable Racah operators.

Abstract

The goal of the present paper is to provide a detailed study of irreducible representations of the algebra generated by the symmetries of the generic quantum superintegrable system on the $d$-sphere. Appropriately normalized, the symmetry operators preserve the space of polynomials. Under mild conditions on the free parameters, maximal abelian subalgebras of the symmetry algebra, generated by Jucys-Murphy elements, have unique common eigenfunctions consisting of families of Jacobi polynomials in $d$ variables. We describe the action of the symmetries on the basis of Jacobi polynomials in terms of multivariable Racah operators, and combine this with different embeddings of symmetry algebras of lower dimensions to prove that the representations restricted on the space of polynomials of a fixed total degree are irreducible.