Matrix Valued Orthogonal Polynomials for Gelfand Pairs of Rank One

2013/10/18 by Gert Heckman, Heckman, Gert, Maarten van Pruijssen +1 · 2 citations
Physics and Astronomy · Mathematics · #Nonlinear Waves and Solitons #Advanced Topics in Algebra #Quantum Mechanics and Non-Hermitian Physics

paper · pdf · doi:10.48550/arxiv.1310.5134

Abstract

In this paper we study matrix valued orthogonal polynomials of one variable associated with a compact connected Gelfand pair (G,K) of rank one, as a generalization of earlier work by Koornwinder and subsequently by Koelink, van Pruijssen and Roman for the pair (SU(2) x SU(2),SU(2)), and by Grünbaum, Pacharoni and Tirao for the pair (SU(3),U(2)). Our method is based on representation theory using an explicit determination of the relevant branching rules. Our matrix valued orthogonal polynomials have the Sturm--Liouville property of being eigenfunctions of a second order matrix valued linear differential operator coming from the Casimir operator, and in fact are eigenfunctions of a commutative algebra of matrix valued linear differential operators coming from U(g_c)^K.

Citations

• Über Sturm-Liouvillesche Polynomsysteme

Cited by

• Algebraic interpretation of discrete families of matrix valued orthogonal polynomials
• Branching rules for finite-dimensional U_q(\mathfraksu(3))-representations with respect to a right coideal subalgebra

Related

• Deformation of matrix-valued orthogonal polynomials related to Gelfand pairs
• Explicit construction of matrix-valued orthogonal polynomials of arbitrary size
• Second order differential operators having several families of orthogonal matrix polynomials as eigenfunctions
• Matrix elements of irreducible representations of SU(n+1)\timesSU(n+1) and multivariable matrix-valued orthogonal polynomials
• Matrix orthogonal polynomials whose derivatives are also orthogonal