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