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

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Nicolas Crampé , Julien Gaboriaud , Loïc Poulain d’Andecy , Luc Vinet ·

YOU? · · 2023 · Open Access · · DOI: https://doi.org/10.48550/arxiv.2308.12809

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.

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Mathematics Algebra over a field Pure mathematics Kravchuk polynomials Irreducibility Centralizer and normalizer Orthogonal polynomials Recurrence relation Irreducible representation Classical orthogonal polynomials Gegenbauer polynomials Combinatorics

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Type: preprint
Language: en
Landing Page: http://arxiv.org/abs/2308.12809
PDF: https://arxiv.org/pdf/2308.12809
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References: 57
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DOI: https://doi.org/10.48550/arxiv.2308.12809

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

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Publication year: 2023

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Authors: Nicolas Crampé, Julien Gaboriaud, Loïc Poulain d’Andecy, Luc Vinet

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Concepts: Mathematics, Algebra over a field, Pure mathematics, Kravchuk polynomials, Irreducibility, Centralizer and normalizer, Orthogonal polynomials, Recurrence relation, Irreducible representation, Classical orthogonal polynomials, Gegenbauer polynomials, Combinatorics

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