Algebraic interpretation of discrete families of matrix valued orthogonal polynomials

2025/10/29 by Labriet, Quentin, Morey, Lucia, Vinet, Luc
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2510.25948

Abstract

An algebraic interpretation of matrix-valued orthogonal polynomials (MVOPs) is provided. The construction is based on representations of a (q-deformed) Lie algebra \mathfrakg into the algebra End_Mn(ℂ)(M) of M_n(ℂ)-linear maps over a M_n(ℂ)-module M. Cases corresponding to the Lie algebras \mathfraksu(2) and \mathfraksu(1, 1) as well as to the q-deformed algebra \mathfrakso_q(3) at q a root of unity are presented; they lead to matrix analogs of the Krawtchouk, Meixner and discrete Chebyshev polynomials.

Citations

• Matrix-Valued Orthogonal Polynomials Related to a Class of Random Walk
• An evolution of matrix-valued orthogonal polynomials
• Duality and difference operators for matrix valued discrete polynomials on the nonnegative integers
• Matrix Valued Orthogonal Polynomials for Gelfand Pairs of Rank One
• Matrix Valued Spherical Functions Associated to the Complex Projective Plane
• Orthogonal matrix polynomials and higher order recurrence relations

Related

• Meixner d-Orthogonal Polynomials Arising from \mathfraksu(1,1)
• Constructing bispectral orthogonal polynomials from the classical discrete families of Charlier, Meixner and Krawtchouk
• Explicit construction of matrix-valued orthogonal polynomials of arbitrary size
• Structure of operator algebras for matrix orthogonal polynomials
• Automorphisms of algebras and Bochner`s property for discrete vector orthogonal polynomials