Matrix Valued Spherical Functions Associated to the Complex Projective Plane

Grünbaum, F. Alberto; Pacharoni, I.; Tirao, Juan

doi:10.1006/jfan.2001.3840

Cited by 124 publications

(162 citation statements)

References 17 publications

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“…Typically they are joint eigenfunctions of some fixed differential operator with matrix coefficients. This search was initiated in [5], but nontrivial examples were not discovered until [6] and [13,16]. The family of examples we will consider later is related to one of these examples and is obtained by modifying the situation discussed in [30].…”

Section: Now For I ≥ 0 We Havementioning

confidence: 99%

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Matrix Valued Orthogonal Polynomials Arising from Group Representation Theory and a Family of Quasi-Birth-and-Death Processes

Grünbaum¹,

Iglesia²

2008

SIAM J. Matrix Anal. & Appl.

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Abstract. We consider a family of matrix valued orthogonal polynomials obtained by Pacharoni and Tirao in connection with spherical functions for the pair (SU(N + 1), U(N )); see [I. Pacharoni and J. A. Tirao, Constr. Approx., 25 (2007), pp. 177-192]. After an appropriate conjugation, we obtain a new family of matrix valued orthogonal polynomials where the corresponding block Jacobi matrix is stochastic and has special probabilistic properties. This gives a highly nontrivial example of a nonhomogeneous quasi-birth-and-death process for which we can explicitly compute its "nstep transition probability matrix" and its invariant distribution. The richness of the mathematical structures involved here allows us to give these explicit results for a several parameter family of quasi-birth-and-death processes with an arbitrary (finite) number of phases. Some of these results are plotted to show the effect that choices of the parameter values have on the invariant distribution.

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Section: Now For I ≥ 0 We Havementioning

confidence: 99%

“…The first family of examples appeared in [13,16] in connection with G = SU(3). The size of the matrices here is already arbitrary, and the orthogonality matrix has a scalar factor of the form x α (1 − x).…”

Section: Matrix Valued Spherical Functions and Matrix Valued Orthogonmentioning

confidence: 99%

Matrix Valued Orthogonal Polynomials Arising from Group Representation Theory and a Family of Quasi-Birth-and-Death Processes

Grünbaum¹,

Iglesia²

2008

SIAM J. Matrix Anal. & Appl.

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“…More generally, it is well known that the zonal spherical functions associated to real compact symmetric spaces can be realized as Jacobi polynomials. The link between zonal spherical functions and orthogonal polynomials has a matrix-valued analogue that was first investigated in [11] for the compact symmetric pair (G, K) = (SU(3), U(2)). The matrix-valued spherical functions are related to an auxiliary function which is an eigenfunction of a matrix-valued differential operator related to the Casimir operator of the group G and that is given explicitly.…”

Section: Introductionmentioning

confidence: 99%

Some bivariate stochastic models arising from group representation theory

Iglesia

Román

2018

Stochastic Processes and their Applications

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Abstract. The aim of this paper is to study some continuous-time bivariate Markov processes arising from group representation theory. The first component (level) can be either discrete (quasi-birth-and-death processes) or continuous (switching diffusion processes), while the second component (phase) will always be discrete and finite. The infinitesimal operators of these processes will be now matrix-valued (either a block tridiagonal matrix or a matrix-valued secondorder differential operator). The matrix-valued spherical functions associated to the compact symmetric pair (SU(2) × SU(2), diag SU(2)) will be eigenfunctions of these infinitesimal operators, so we can perform spectral analysis and study directly some probabilistic aspects of these processes. Among the models we study there will be rational extensions of the one-server queue and Wright-Fisher models involving only mutation effects.

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“…In this paper, it is reduced to the Clebsch-Gordan decomposition, and there is a nice result by Oblomkov and Stokman [38,Proposition 1.15] on a special case of the branching rule for quantum symmetric pair of type AIII, but in general the lack of the branching rule for the quantum symmetric pairs is an obstacle for the study of quantum analogues of matrix-valued spherical functions of e.g. [17,18,38,44].…”

Section: Introductionmentioning

confidence: 99%

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

2016

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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

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Matrix Valued Spherical Functions Associated to the Complex Projective Plane

Cited by 124 publications

References 17 publications

Matrix Valued Orthogonal Polynomials Arising from Group Representation Theory and a Family of Quasi-Birth-and-Death Processes

Matrix Valued Orthogonal Polynomials Arising from Group Representation Theory and a Family of Quasi-Birth-and-Death Processes

Some bivariate stochastic models arising from group representation theory

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

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