Matrix elements of $SO(3)$ in $s{l}_3$ representations as bispectral multivariate functions

Abstract

We compute the matrix elements of $SO(3)$ in any finite-dimensional irreducible representation of $s{l}_3$. 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 $s{l}_2$ in $s{l}_3$. 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 $s{l}_2$ Casimir elements related to the two embeddings of $s{l}_2$ in $s{l}_3$ obey the Racah algebra relations. We also show that these two elements generate the centralizer in $U(s{l}_3)$ of the Cartan subalgebra and its complete algebraic description is given.

arXiv comment20 pages. v2 updated to match the version accepted for publication

References in corpus (25)

• Superintegrability in two dimensions and the Racah-Wilson algebra
• Bispectrality of multivariable Racah-Wilson polynomials
• The Relationship between Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case
• A higher rank Racah algebra and the ${\mathbb{Z}}_2^n$ Laplace-Dunkl operator
• The multivariate Krawtchouk polynomials as matrix elements of the rotation group representations on oscillator states
• Two-Variable Wilson Polynomials and the Generic Superintegrable System on the 3-Sphere
• A Probablistic Origin for a New Class of Bivariate Polynomials
• A Lie theoretic interpretation of multivariate hypergeometric polynomials
• The equitable Racah algebra from three su(1,1) algebras
• The Rahman polynomials and the Lie algebra sl_3(C)
• The Racah algebra and superintegrable models
• The Askey-Wilson algebra and its avatars
• Representations of the rank two Racah algebra and orthogonal multivariate polynomials
• Embeddings of the Racah Algebra into the Bannai-Ito Algebra
• Racah Polynomials and Recoupling Schemes of $\mathfrak{su}(1,1)$
• The multivariate Charlier polynomials as matrix elements of the Euclidean group representation on oscillator states
• Finite-Dimensional Irreducible Modules of the Racah Algebra at Characteristic Zero
• Racah algebras, the centralizer $Z_n({\mathfrak{sl}}_2)$ and its Hilbert-Poincaré series
• Algebraic (super-)integrability from commutants of subalgebras in universal enveloping algebras
• The Racah algebra as a commutant and Howe duality
• Polynomial algebras from $su(3)$ and the generic model on the two sphere
• Gelfand-Tsetlin bases for classical Lie algebras
• Racah problems for the oscillator algebra, the Lie algebra ${\mathfrak{sl}}_n$, and multivariate Krawtchouk polynomials
• $q$-Rotations and Krawtchouk polynomials
• New realizations of algebras of the Askey-Wilson type in terms of Lie and quantum algebras