Matrix-valued spherical functions related to the quantum symmetric pair for the quantum analogue of (SU(2) × SU(2), 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, and 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.
B Pablo Román
Matrix-valued analogues of the little q-Jacobi polynomials are introduced and studied. For the 2×2-matrix-valued little q-Jacobi polynomials explicit expressions for the orthogonality relations, Rodrigues formula, three-term recurrence relation and their relation to matrix-valued q-hypergeometric series and the scalar-valued little q-Jacobi polynomials are presented. The study is based on a matrix-valued q-difference operator, which is a q-analogue of Tirao's matrix-valued hypergeometric differential operator.
For a one-parameter family of lower triangular matrices with entries involving continuous qultraspherical polynomials we give an explicit lower triangular inverse matrix, with entries involving again continuous q-ultraspherical functions. The matrices are q-analogues of results given by Cagliero and Koornwinder recently. The proofs are not q-analogues of the Cagliero-Koornwinder case, but are of a different nature involving q-Racah polynomials. Some applications of these new formulas are given. Also the limit β → 0 is studied and gives rise to continuous q-Hermite polynomials for 0 < q < 1 and q > 1.
We consider the quantum symmetric pair (U q (su(3)), B) where B is a right coideal subalgebra. We prove that all finite-dimensional irreducible representations of B are weight representations and are characterised by their highest weight and dimension. We show that the restriction of a finite-dimensional irreducible representation of U q (su(3)) to B decomposes multiplicity free into irreducible representations of B. Furthermore we give explicit expressions for the highest weight vectors in this decomposition in terms of dual q-Krawtchouk polynomials.
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