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

Abstract

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.

28 pages