Representations of the rank two Racah algebra and orthogonal multivariate polynomials

Abstract

The algebraic structure of the rank two Racah algebra is studied in detail. We provide an automorphism group of this algebra, which is isomorphic to the permutation group of five elements. This group can be geometrically interpreted as the symmetry of a folded icosidodecahedron. It allows us to study a class of equivalent irreducible representations of this Racah algebra. They can be chosen symmetric so that their transition matrices are orthogonal. We show that their entries can be expressed in terms of Racah polynomials. This construction gives an alternative proof of the recurrence, difference and orthogonal relations satisfied by the Tratnik polynomials, as well as their expressions as a product of two monovariate Racah polynomials. Our construction provides a generalization of these bivariate polynomials together with their properties.

40 pages, 4 figures. Several signs corrected and some proofs clarified

References in corpus (5)

• The Universal Askey-Wilson Algebra
• Two-Variable Wilson Polynomials and the Generic Superintegrable System on the 3-Sphere
• Models of Quadratic Algebras Generated by Superintegrable Systems in 2D
• Embeddings of the Racah Algebra into the Bannai-Ito Algebra
• Racah algebras, the centralizer $Z_n({\mathfrak{sl}}_2)$ and its Hilbert-Poincaré series

Cited by in corpus (8)

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• An Askey-Wilson Algebra of Rank 2
• Gaudin model for the multinomial distribution
• Polynomial algebras from commutants: Classical and Quantum aspects of ${\mathcal{A}}_3$
• $\lambda$-Griffiths polynomials: Bispectrality and biorthogonality
• Matrix elements of $SO(3)$ in $s{l}_3$ representations as bispectral multivariate functions
• Factorized $A_2$-Leonard pair