  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.
