Matrix elements of $SO(3)$ in $sl_3$ representations as bispectral multivariate functions

TL;DR

This paper computes matrix elements of $SO(3)$ in $sl_3$ representations, expressing them via multivariate polynomials, and explores their algebraic and recurrence properties, revealing connections to Racah algebra and bispectral functions.

Contribution

It provides explicit formulas for $SO(3)$ matrix elements in $sl_3$ representations using multivariate Krawtchouk and Racah polynomials, and analyzes their algebraic structure.

Findings

Matrix elements expressed as double sums of Krawtchouk and Racah polynomials.

Recurrence and difference relations derived for these polynomials.

Identification of Racah algebra relations in the context of $sl_3$ representations.

Abstract

We compute the matrix elements of $SO(3)$ in any finite-dimensional irreducible representation of $sl_3$. They are expressed in terms of a double sum of products of Krawtchouk and Racah polynomials which generalize the Griffiths-Krawtchouk polynomials. Their recurrence and difference relations are obtained as byproducts of our construction. The proof is based on the decomposition of a general three-dimensional rotation in terms of elementary planar rotations and a transition between two embeddings of $sl_2$ in $sl_3$. The former is related to monovariate Krawtchouk polynomials and the latter, to monovariate Racah polynomials. The appearance of Racah polynomials in this context is algebraically explained by showing that the two $sl_2$ Casimir elements related to the two embeddings of $sl_2$ in $sl_3$ obey the Racah algebra relations. We also show that these two elements generate the centralizer in $U(sl_3)$ of the Cartan subalgebra and its complete algebraic description is given.