Symmetry algebra for the generic superintegrable system on the sphere

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.

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Cited by in corpus (10)

• Higher Order Quantum Superintegrability: a new "Painlevé conjecture"
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• New superintegrable models on spaces of constant curvature
• Bargmann and Barut-Girardello models for the Racah algebra
• Gaudin model for the multinomial distribution
• A discrete realization of the higher rank Racah algebra
• Darboux transformations from the Appell-Lauricella operator
• Non-Hermitian superintegrable systems
• The Racah algebra and ${\mathfrak{sl}}_n$