Multiplicity Free Induced Representations and Orthogonal Polynomials

Pruijssen, Maarten van

doi:10.1093/imrn/rnw295

Cited by 10 publications

(25 citation statements)

References 35 publications

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“…General nonorthogonal decompositions are governed by the real vector space [22,44]. In [28] we show that A (ν) equals the Hermitian elements of A (ν) , so that there is no further nonorthogonal decomposition.…”

Section: Proposition 26mentioning

confidence: 92%

Matrix-Valued Gegenbauer-Type Polynomials

2017

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We introduce matrix-valued weight functions of arbitrary size, which are analogues of the weight function for the Gegenbauer or ultraspherical polynomials for the parameter ν > 0. The LDU-decomposition of the weight is explicitly given in terms of Gegenbauer polynomials. We establish a matrix-valued Pearson equation for these matrix weights leading to explicit shift operators relating the weights for parameters ν and ν + 1. The matrix coefficients of the Pearson equation are obtained using a special matrix-valued differential operator in a commutative algebra of symmetric differential operators. The corresponding orthogonal polynomials Approx (2017) 46:459-487 are the matrix-valued Gegenbauer-type polynomials which are eigenfunctions of the symmetric matrix-valued differential operators. Using the shift operators, we find the squared norm, and we establish a simple Rodrigues formula. The three-term recurrence relation is obtained explicitly using the shift operators as well. We give an explicit nontrivial expression for the matrix entries of the matrix-valued Gegenbauertype polynomials in terms of scalar-valued Gegenbauer and Racah polynomials using the LDU-decomposition and differential operators. The case ν = 1 reduces to the case of matrix-valued Chebyshev polynomials previously obtained using group theoretic considerations.

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Section: Proposition 26mentioning

confidence: 92%

Matrix-Valued Gegenbauer-Type Polynomials

2017

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“…A first step was made by Koornwinder [34] realizing vector-valued orthogonal polynomials, and next Grünbaum, Pacharoni, Tirao [20] studied matrix valued orthogonal polynomials related to the symmetric pair (SU(3), U(2)) using differential operators. These papers have been motivational to understand the situation of matrix valued orthogonal polynomials in connection to representation theory, and in particular we refer to [1], [23], [29], [30], [31], [38], [39].…”

Section: Introductionmentioning

confidence: 99%

Matrix Valued Laguerre Polynomials

Koelink

Román

2019

Trends in Mathematics

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Matrix valued Laguerre polynomials are introduced via a matrix weight function involving several degrees of freedom using the matrix nature. Under suitable conditions on the parameters the matrix weight function satisfies matrix Pearson equations, which allow to introduce shift operators for these polynomials. The shift operators lead to explicit expressions for the structures of these matrix valued Laguerre polynomials, such as a Rodrigues formula, the coefficients in the three-term recurrence, differential operators, and expansion formulas.

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“…Results. One of the main results in [39] is the existence of families of multivariable matrix-valued orthogonal polynomials that are simultaneous eigenfunctions of a commutative algebra of differential operators. The existence is based on examples and an ad hoc analysis of the involved spectra.…”

Section: Motivation and Historymentioning

confidence: 99%

“…We recall some of the results of [39] specified to symmetric spaces. The main idea, to view the space spanned by the spherical functions as a module over the ring of biinvariant functions, originates from [21,38] and is based on the classical results in [49].…”

Section: Matrix-valued Spherical Functionsmentioning

confidence: 99%

“…There is an abundance of examples of multiplicity free triples, namely those coming from the multiplicity free systems, i.e. triples (G, H, P ) with (G, H) as before and with P ⊂ H a parabolic subgroup such that G/P admits an open orbit of a Borel subgroup of G. Any positive character µ ∈ X + (T H ) that extends to a character µ : P → C × gives rise to a multiplicity free triple, see [39].…”

Section: Matrix-valued Spherical Functionsmentioning

confidence: 99%

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Matrix elements of irreducible representations of SU(n + 1)×SU(n + 1) and multivariable matrix-valued orthogonal polynomials

Koelink

Pruijssen

Román

2020

Journal of Functional Analysis

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In Part 1 we study the spherical functions on compact symmetric pairs of arbitrary rank under a suitable multiplicity freeness assumption and additional conditions on the branching rules. The spherical functions are taking values in the spaces of linear operators of a finite dimensional representation of the subgroup, so the spherical functions are matrix-valued. Under these assumptions these functions can be described in terms of matrix-valued orthogonal polynomials in several variables, where the number of variables is the rank of the compact symmetric pair. Moreover, these polynomials are uniquely determined as simultaneous eigenfunctions of a commutative algebra of differential operators.In Part 2 we verify that the group case SU(n + 1) meets all the conditions that we impose in Part 1. For any k ∈ N 0 we obtain families of orthogonal polynomials in n variables with values in the N × N -matrices, where N = n+k k . The case k = 0 leads to the classical Heckman-Opdam polynomials of type A n with geometric parameter. For k = 1 we obtain the most complete results. In this case we give an explicit expression of the matrix weight, which we show to be irreducible whenever n ≥ 2. We also give explicit expressions of the spherical functions that determine the matrix weight for k = 1. These expressions are used to calculate the spherical functions that determine the matrix weight for general k up to invertible upper-triangular matrices. This generalizes and gives a new proof of a formula originally obtained by Koornwinder for the case n = 1. The commuting family of differential operators that have the matrix-valued polynomials as simultaneous eigenfunctions contains an element of order one. We give explicit formulas for differential operators of order one and two for (n, k) equal to (2, 1) and (3, 1). 4 5

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Multiplicity Free Induced Representations and Orthogonal Polynomials

Cited by 10 publications

References 35 publications

Matrix-Valued Gegenbauer-Type Polynomials

Matrix-Valued Gegenbauer-Type Polynomials

Matrix Valued Laguerre Polynomials

Matrix elements of irreducible representations of SU(n + 1)×SU(n + 1) and multivariable matrix-valued orthogonal polynomials

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Multiplicity Free Induced Representations and Orthogonal Polynomials

Cited by 10 publications

References 35 publications

Matrix-Valued Gegenbauer-Type Polynomials

Matrix-Valued Gegenbauer-Type Polynomials

Matrix Valued Laguerre Polynomials

Matrix elements of irreducible representations of SU(n + 1)×SU(n + 1) and multivariable matrix-valued orthogonal polynomials

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About

Matrix elements of irreducible representations of SU(n + 1)×SU(n + 1) and multivariable matrix-valued orthogonal polynomials