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We compute the matrix elements of $SO(3)$ in any finite-dimensionalirreducible representation of $sl_3$. They are expressed in terms of a doublesum of products of Krawtchouk and Racah polynomials which generalize theGriffiths-Krawtchouk polynomials. Their recurrence and difference relations areobtained as byproducts of our construction. The proof is based on thedecomposition of a general three-dimensional rotation in terms of elementaryplanar rotations and a transition between two embeddings of $sl_2$ in $sl_3$.The former is related to monovariate Krawtchouk polynomials and the latter, tomonovariate Racah polynomials. The appearance of Racah polynomials in thiscontext is algebraically explained by showing that the two $sl_2$ Casimirelements related to the two embeddings of $sl_2$ in $sl_3$ obey the Racahalgebra relations. We also show that these two elements generate thecentralizer in $U(sl_3)$ of the Cartan subalgebra and its complete algebraicdescription is given.

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