Matrix-valued orthogonal polynomials related to the quantum analogue of $(SU(2) \times SU(2), \text{diag})$

TL;DR

This paper introduces and analyzes matrix-valued orthogonal polynomials arising from quantum symmetric pairs related to the quantum analogue of (SU(2)×SU(2), diag), deriving their properties using quantum group techniques.

Contribution

It constructs matrix-valued orthogonal polynomials from quantum symmetric pairs and derives their orthogonality, recurrence, and explicit formulas using quantum group methods.

Findings

Derived orthogonality relations from Schur orthogonality

Established three-term recurrence relations

Provided explicit formulas for matrix entries using special polynomials

Abstract

Matrix-valued spherical functions related to the quantum symmetric pair for the quantum analogue of $(SU(2) \times SU(2), \text{diag})$ are introduced and studied in detail. The quantum symmetric pair is given in terms of a quantised universal enveloping algebra with a coideal subalgebra. The matrix-valued spherical functions give rise to matrix-valued orthogonal polynomials, which are matrix-valued analogues of a subfamily of Askey-Wilson polynomials. For these matrix-valued orthogonal polynomials a number of properties are derived using this quantum group interpretation: the orthogonality relations from the Schur orthogonality relations, the three-term recurrence relation and the structure of the weight matrix in terms of Chebyshev polynomials from tensor product decompositions, the matrix-valued Askey-Wilson type $q$-difference operators from the action of the Casimir elements. A more analytic study of the weight gives an explicit LDU-decomposition in terms of continuous $q$-ultraspherical polynomials. The LDU-decomposition gives the possibility to find explicit expressions of the matrix entries of the matrix-valued orthogonal polynomials in terms of continuous $q$-ultraspherical polynomials and $q$-Racah polynomials.