Racah algebras, the centralizer $Z_n(\mathfrak{sl}_2)$ and its Hilbert-Poincar\'e series

The higher rank Racah algebra $R(n)$ introduced recently 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 $Z_{n}(\mathfrak{sl}_2)$ of the diagonal embedding of $U(\mathfrak{sl}_2)$ in $U(\mathfrak{sl}_2)^{\otimes n}$. This leads to a first and novel presentation of the centralizer $Z_{n}(\mathfrak{sl}_2)$ in terms of generators and defining relations. An explicit formula of its Hilbert-Poincar\'e 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.