Matrix-Valued Orthogonal Polynomials Related to (SU(2)×SU(2), diag)

Abstract

In a previous paper we have introduced matrix-valued analogues of the Chebyshev polynomials by studying matrix-valued spherical functions on SU(2) × SU(2). In particular the matrix-size of the polynomials is arbitrarily large. The matrix-valued orthogonal polynomials and the corresponding weight function are studied. In particular, we calculate the LDU-decomposition of the weight where the matrix entries of L are given in terms of Gegenbauer polynomials. The monic matrix-valued orthogonal polynomials P n are expressed in terms of Tirao's matrix-valued hypergeometric function using the matrix-valued differential operator of first and second order to which the P n 's are eigenfunctions. From this result we obtain an explicit formula for coefficients in the three-term recurrence relation satisfied by the polynomials P n . These differential operators are also crucial in expressing the matrix entries of P n L as a product of a Racah and a Gegenbauer polynomial. We also present a group theoretic derivation of the matrix-valued differential operators by considering the Casimir operators corresponding to SU(2) × SU(2).

Some bivariate stochastic models arising from group representation theory

“…Here we discuss the family of matrix-valued spherical functions given in [26,27] for the pair (G, K) = (SU(2)×SU(2), diag SU(2)) and the one-parameter extension [25,37]. For each ∈ N, if we let N = 2 + 1, it was shown in [26,27] that there exists a family of C N ×N -valued functions {Ψ n : n ∈ N 0 }, defined on the interval [0, 1]. The family is constructed by means of the spherical functions associated to (G, K).…”

“…A probabilistic interpretation for this case is given in [10] and is extended in [12]. An alternative approach to relate matrix-valued spherical functions and matrix-valued orthogonal polynomials is given in [26,27,15,37], where more general families of symmetric pairs (G, K) are treated. In this construction, one obtains a family of matrix-valued functions Ψ n , together with a matrix-valued differential operator Ω, for which the functions Ψ n are eigenfunctions.…”

“…In Section 2 we will give a brief account of matrix-valued spherical functions, focusing on the example for the pair (G, K) = (SU(2) × SU(2), diag SU(2)) studied in [26,27] and the one-parameter extension given in [25,37]. The second-order differential operator, three-term recurrence relation, weight matrix, norms and other structural formulas to transform the operators into operators with stochastic interpretation will be given.…”

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

“…However, in order to determine the matrix coefficients in the matrix-valued three-term recurrence we use analytic methods. The existence of two Casimir elements in U q (g) leads to the matrix-valued orthogonal polynomials being eigenfunctions of two commuting matrix-valued q-difference operators, see [23] for the group case. This extends Letzter [35] to the matrix-valued set-up for this particular case.…”

Branching Rules for Finite-Dimensional U q ( 𝖘 𝖚 ( 3 ) ) $\mathcal {U}_{q}(\mathfrak {su}(3))$ -Representations with Respect to a Right Coideal Subalgebra

“…The quantum symmetric pair is given by the quantised universal enveloping algebra of U q (g), where g = su(2) ⊕ su (2), and a right coideal subalgebra B that can be identified with U q (su (2)). As in the Lie group setting [8,11,12,27], the explicit knowledge of the branching rules plays a fundamental role in the explicit determination of the matrix-valued spherical functions. In this first case, the branching rules for the irreducible representations of U q (g) with respect to B follow using the standard Clebsch-Gordan decomposition.…”