Racah Problems for the Oscillator Algebra, the Lie Algebra $$\mathfrak {sl}_n$$ sl n , and Multivariate Krawtchouk Polynomials

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.