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.
Skip Nav Destination
Article navigation
November 2023
Research Article|
November 30 2023
Matrix elements of SO(3) in sl3 representations as bispectral multivariate functions
Nicolas Crampé
;
Nicolas Crampé
a)
(Conceptualization, Investigation, Writing – original draft, Writing – review & editing)
1
Institut Denis-Poisson CNRS/UMR 7013 - Université de Tours - Université d’Orléans
, Parc de Grandmont, 37200 Tours, France
Search for other works by this author on:
Julien Gaboriaud
;
Julien Gaboriaud
b)
(Conceptualization, Investigation, Writing – original draft, Writing – review & editing)
2
Graduate School of Informatics, Kyoto University
, Sakyo-ku, Kyoto 606-8501, Japan
b)Author to whom correspondence should be addressed: [email protected]
Search for other works by this author on:
Loïc Poulain d’Andecy
;
Loïc Poulain d’Andecy
c)
(Conceptualization, Investigation, Writing – original draft, Writing – review & editing)
3
Laboratoire de mathématiques de Reims UMR 9008, Université de Reims Champagne-Ardenne
, Moulin de la Housse BP 1039, 51100 Reims, France
Search for other works by this author on:
Luc Vinet
Luc Vinet
d)
(Conceptualization, Investigation, Writing – original draft, Writing – review & editing)
4
Centre de Recherches Mathématiques, Université de Montréal
, P.O. Box 6128, Centre-ville Station, Montréal, Québec H3C 3J7, Canada
5
IVADO
, Montréal, Québec H2S 3H1, Canada
Search for other works by this author on:
b)Author to whom correspondence should be addressed: [email protected]
a)
E-mail: [email protected]
c)
E-mail: [email protected]
d)
E-mail: [email protected]
J. Math. Phys. 64, 111702 (2023)
Article history
Received:
August 24 2023
Accepted:
October 03 2023
Citation
Nicolas Crampé, Julien Gaboriaud, Loïc Poulain d’Andecy, Luc Vinet; Matrix elements of SO(3) in sl3 representations as bispectral multivariate functions. J. Math. Phys. 1 November 2023; 64 (11): 111702. https://doi.org/10.1063/5.0173787
Download citation file:
Pay-Per-View Access
$40.00
Sign In
You could not be signed in. Please check your credentials and make sure you have an active account and try again.
15
Views
Citing articles via
Derivation of the Maxwell–Schrödinger equations: A note on the infrared sector of the radiation field
Marco Falconi, Nikolai Leopold
Learning from insulators: New trends in the study of conductivity of metals
Giuseppe De Nittis, Max Lein, et al.
Cascades of scales: Applications and mathematical methodologies
Luigi Delle Site, Rupert Klein, et al.
Related Content
Sklyanin-like algebras for (q-)linear grids and (q-)para-Krawtchouk polynomials
J. Math. Phys. (January 2021)
A bispectral q-hypergeometric basis for a class of quantum integrable models
J. Math. Phys. (January 2018)
SUq(3) corepresentations and bivariate q-Krawtchouk polynomials
J. Math. Phys. (May 2019)
The Heun–Racah and Heun–Bannai–Ito algebras
J. Math. Phys. (August 2020)
Heun operator of Lie type and the modified algebraic Bethe ansatz
J. Math. Phys. (August 2021)