The Rahman polynomials and the Lie algebra sl_3(C)

Abstract

We interpret the Rahman polynomials in terms of the Lie algebra $s{l}_3(C)$. Using the parameters of the polynomials we define two Cartan subalgebras for $s{l}_3(C)$, denoted $H$ and $\tilde{H}$. We display an antiautomorphism $†$ of $s{l}_3(C)$ that fixes each element of $H$ and each element of $\tilde{H}$. We consider a certain finite-dimensional irreducible $s{l}_3(C)$-module $V$ consisting of homogeneous polynomials in three variables. We display a nondegenerate symmetric bilinear form $<,>$ on $V$ such that $<\beta \xi ,\zeta >=<\xi ,{\beta }^{†}\zeta >$ for all $\beta \in s{l}_3(C)$ and $\xi ,\zeta \in V$. We display two bases for $V$; one diagonalizes $H$ and the other diagonalizes $\tilde{H}$. Both bases are orthogonal with respect to $<,>$. We show that when $<,>$ is applied to a vector in each basis, the result is a trivial factor times a Rahman polynomial evaluated at an appropriate argument. Thus for both transition matrices between the bases each entry is described by a Rahman polynomial. From these results we recover the previously known orthogonality relation for the Rahman polynomials. We also obtain two seven-term recurrence relations satisfied by the Rahman polynomials, along with the corresponding relations satisfied by the dual polynomials. These recurrence relations show that the Rahman polynomials are bispectral. In our theory the roles of $H$ and $\tilde{H}$ are interchangable, and for us this explains the duality and bispectrality of the Rahman polynomials. We view the action of $H$ and $\tilde{H}$ on $V$ as a rank 2 generalization of a Leonard pair.

Cited by in corpus (15)

• The multivariate Krawtchouk polynomials as matrix elements of the rotation group representations on oscillator states
• A System of Multivariable Krawtchouk Polynomials and a Probabilistic Application
• On a Family of 2-Variable Orthogonal Krawtchouk Polynomials
• Entanglement of inhomogeneous free fermions on hyperplane lattices
• A superintegrable finite oscillator in two dimensions with SU(2) symmetry
• Racah problems for the oscillator algebra, the Lie algebra ${\mathfrak{sl}}_n$, and multivariate Krawtchouk polynomials
• Bivariate $P$-polynomial association schemes
• Spin lattices, state transfer and bivariate Krawtchouk polynomials
• $S{U}_q(3)$ corepresentations and bivariate $q$-Krawtchouk polynomials
• Quantum Walks on Graphs of the Ordered Hamming Scheme and Spin Networks
• An Assmus-Mattson theorem for codes over commutative association schemes
• $\lambda$-Griffiths polynomials: Bispectrality and biorthogonality
• Factorized $A_2$-Leonard pair
• Matrix elements of $SO(3)$ in $s{l}_3$ representations as bispectral multivariate functions
• Rahman polynomials