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- W4386185445 abstract "We consider the change of basis under an $SO(3)$ automorphism for a finite-dimensional irreducible representation of $sl_3$. The coefficients of the transition matrix are expressed in terms of a double sum of products of Krawtchouk and Racah polynomials. These functions generalize the Griffiths-Krawtchouk polynomials, which are essentially the overlap coefficients in the case of symmetric representations of $sl_3$. 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." @default.
- W4386185445 created "2023-08-26" @default.
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- W4386185445 date "2023-08-24" @default.
- W4386185445 modified "2023-09-28" @default.
- W4386185445 title "Matrix elements of $SO(3)$ in $sl_3$ representations as bispectral multivariate functions" @default.
- W4386185445 doi "https://doi.org/10.48550/arxiv.2308.12809" @default.
- W4386185445 hasPublicationYear "2023" @default.
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