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The oscillator Racah algebra $\mathcal{R}_n(\mathfrak{h})$ is realized by theintermediate Casimir operators arising in the multifold tensor product of theoscillator algebra $\mathfrak{h}$. An embedding of the Lie algebra$\mathfrak{sl}_{n-1}$ into $\mathcal{R}_n(\mathfrak{h})$ is presented. Itrelates the representation theory of the two algebras. We establish theconnection between recoupling coefficients for $\mathfrak{h}$ and matrixelements of $\mathfrak{sl}_n$-representations which are both expressed in termsof multivariate Krawtchouk polynomials of Griffiths type.

Racah problems for the oscillator algebra, the Lie algebra $\mathfrak{sl}_n$, and multivariate Krawtchouk polynomials