Matrix elements of SO(3) in sl3 representations as bispectral multivariate functions

We compute the matrix elements of SO(3) in any finite-dimensional irreducible representation of sl3. 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 obtain...

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Bibliographic Details
Published in:Journal of mathematical physics 2023-11, Vol.64 (11)
Main Authors: Crampé, Nicolas, Gaboriaud, Julien, Poulain d’Andecy, Loïc, Vinet, Luc
Format: Article
Language:English
Subjects:
Mathematical analysis
Matrices (mathematics)
Matrix algebra
Polynomials
Representations
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Summary:We compute the matrix elements of SO(3) in any finite-dimensional irreducible representation of sl3. 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 sl2 in sl3. 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 sl2 Casimir elements related to the two embeddings of sl2 in sl3 obey the Racah algebra relations. We also show that these two elements generate the centralizer in U(sl3) of the Cartan subalgebra and its complete algebraic description is given.
ISSN:0022-2488
1089-7658
DOI:10.1063/5.0173787