Representations of the rank two Racah algebra and orthogonal multivariate polynomials

TL;DR

This paper explores the algebraic structure of the rank two Racah algebra, revealing its symmetries, representations, and connections to multivariate Racah and Tratnik polynomials, thereby generalizing these polynomials and their properties.

Contribution

It provides a detailed analysis of the rank two Racah algebra's automorphisms, representations, and their relation to multivariate orthogonal polynomials, offering new proofs and generalizations.

Findings

Automorphism group isomorphic to permutation group of five elements

Transition matrices are orthogonal and expressed via Racah polynomials

Generalization of bivariate Racah and Tratnik 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.