Representations of the rank two Racah algebra and orthogonal multivariate polynomials

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 thepermutation group of five elements. This group can be geometrically interpretedas the symmetry of a folded icosidodecahedron. It allows us to study a class ofequivalent irreducible representations of this Racah algebra. They can bechosen symmetric so that their transition matrices are orthogonal. We show thattheir entries can be expressed in terms of Racah polynomials. This constructiongives an alternative proof of the recurrence, difference and orthogonalrelations satisfied by the Tratnik polynomials, as well as their expressions asa product of two monovariate Racah polynomials. Our construction provides ageneralization of these bivariate polynomials together with their properties