Racah Algebras, the Centralizer $$Z_n({{{\mathfrak {s}}}{{\mathfrak {l}}}}_2)$$ and Its Hilbert–Poincaré Series

The higher rank Racah algebra R(n) introduced in [1] is recalled. A quotient of this algebra by central elements, which we call the special Racah algebra sR(n), is then introduced. Using results from classical invariant theory, this sR(n) algebra is shown to be isomorphic to the centralizer Z n (sl 2) of the diagonal embedding of U (sl 2) in U (sl 2) ⊗n. This leads to a first and novel presentation of the centralizer Z n (sl 2) in terms of generators and defining relations. An explicit formula of its Hilbert-Poincaré series is also obtained and studied. The extension of the results to the study of the special Askey-Wilson algebra and its higher rank generalizations is discussed.