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.

arXiv comment28 pages

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Cited by in corpus (5)

• Racah algebras, the centralizer $Z_n({\mathfrak{sl}}_2)$ and its Hilbert-Poincaré series
• Racah Algebra $R(n)$ from Coalgebraic Structures and Chains of $R(3)$ Substructures
• Transposed Poisson structures on solvable and perfect Lie algebras
• Gaudin model for the multinomial distribution
• Matrix elements of $SO(3)$ in $s{l}_3$ representations as bispectral multivariate functions