Multivariable Orthogonal Polynomials as Coupling Coefficients for Lie and Quantum Algebra Representations
(1999) Abstract
 The main topic of the thesis is the connection between representation theory and special functions. We study matrix elements, coupling coefficient, and recoupling coefficients for the simplest Lie and quantum groups.
We show that a large number of multivariable orthogonal and biorthogonal polynomials occurring in the literature may be obtained as coupling coefficients (generalized ClebschGordan coefficients) for multiple tensor products of highest weight representations of the group SU(1,1). In many cases such polynomials have appeared in applications (physics and statistics), and they are also connected with sperical harmonics. The algebraic interpretation yields a simple and unified approach to the study of these... (More)  The main topic of the thesis is the connection between representation theory and special functions. We study matrix elements, coupling coefficient, and recoupling coefficients for the simplest Lie and quantum groups.
We show that a large number of multivariable orthogonal and biorthogonal polynomials occurring in the literature may be obtained as coupling coefficients (generalized ClebschGordan coefficients) for multiple tensor products of highest weight representations of the group SU(1,1). In many cases such polynomials have appeared in applications (physics and statistics), and they are also connected with sperical harmonics. The algebraic interpretation yields a simple and unified approach to the study of these polynomials. The corresponding theory can be developed for the group SU(2) and the oscillator group.
Our original motivation came from ``Hankel theory'', more precisely from the higher order Hankel operators introduced by Svante Janson and Jaak Peetre. The Fourier kernels of such operators are ClebschGordan coefficients, and similarly multivariable coupling coefficients are Fourier kernels of certain multilinear forms. We obtain a Schatten class criterion for these higher order Hankel forms.
We give two new proofs of the triple sum formula for Wigner 9<i>j</i>symbols. These are recoupling coefficients for fourfold tensor products, and appear in the theory of angular momentum in quantum mechanics.
We show that general AskeyWilson and <i>q</i>Racah polynomials arise as matrix elements for the SU(1,1) and SU(2) quantum group, respectively. To obtain this interpretation we introduce some new generalized group elements which include the quantum Weyl element as a degenerate case. We also consider coupling coefficients in the quantum group case. These are multivariable generalizations of the <i>q</i>Racah and AskeyWilson polynomials; however, we focus on the more elementary case of multivariable <i>q</i>Hahn polynomials.
We prove a binomial formula for two variables satisfying a quadratic relation. (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/39544
 author
 Rosengren, Hjalmar ^{LU}
 supervisor
 opponent

 Professor Koornwinder, Tom H, KdV Instituut, Universiteit van Amsterdam, Netherlands
 organization
 publishing date
 1999
 type
 Thesis
 publication status
 published
 subject
 keywords
 higher order Hankel form, Wigner symbol, quadratic algebra, ClebschGordan coefficient, quantum group, Lie group, highest weight representation, multivariable orthogonal and biorthogonal polynomials, multivariable hypergeometric and basic hypergeometric functions, algebraisk topologi, Geometry, Geometri, algebraic topology
 pages
 187 pages
 publisher
 Centre for Mathematical Sciences, Lund University
 defense location
 Mathematics Building, SÃ¶lvegatan 18, Room MH:C
 defense date
 19990506 10:15:00
 external identifiers

 other:ISRN: LUNFMA10101999
 ISBN
 9162835165
 language
 English
 LU publication?
 yes
 id
 02400ea0ed404e32abbae34fa1cbc72c (old id 39544)
 date added to LUP
 20160401 17:11:17
 date last changed
 20181121 20:47:18
@phdthesis{02400ea0ed404e32abbae34fa1cbc72c, abstract = {{The main topic of the thesis is the connection between representation theory and special functions. We study matrix elements, coupling coefficient, and recoupling coefficients for the simplest Lie and quantum groups.<br/><br> <br/><br> We show that a large number of multivariable orthogonal and biorthogonal polynomials occurring in the literature may be obtained as coupling coefficients (generalized ClebschGordan coefficients) for multiple tensor products of highest weight representations of the group SU(1,1). In many cases such polynomials have appeared in applications (physics and statistics), and they are also connected with sperical harmonics. The algebraic interpretation yields a simple and unified approach to the study of these polynomials. The corresponding theory can be developed for the group SU(2) and the oscillator group.<br/><br> <br/><br> Our original motivation came from ``Hankel theory'', more precisely from the higher order Hankel operators introduced by Svante Janson and Jaak Peetre. The Fourier kernels of such operators are ClebschGordan coefficients, and similarly multivariable coupling coefficients are Fourier kernels of certain multilinear forms. We obtain a Schatten class criterion for these higher order Hankel forms.<br/><br> <br/><br> We give two new proofs of the triple sum formula for Wigner 9<i>j</i>symbols. These are recoupling coefficients for fourfold tensor products, and appear in the theory of angular momentum in quantum mechanics.<br/><br> <br/><br> We show that general AskeyWilson and <i>q</i>Racah polynomials arise as matrix elements for the SU(1,1) and SU(2) quantum group, respectively. To obtain this interpretation we introduce some new generalized group elements which include the quantum Weyl element as a degenerate case. We also consider coupling coefficients in the quantum group case. These are multivariable generalizations of the <i>q</i>Racah and AskeyWilson polynomials; however, we focus on the more elementary case of multivariable <i>q</i>Hahn polynomials.<br/><br> <br/><br> We prove a binomial formula for two variables satisfying a quadratic relation.}}, author = {{Rosengren, Hjalmar}}, isbn = {{9162835165}}, keywords = {{higher order Hankel form; Wigner symbol; quadratic algebra; ClebschGordan coefficient; quantum group; Lie group; highest weight representation; multivariable orthogonal and biorthogonal polynomials; multivariable hypergeometric and basic hypergeometric functions; algebraisk topologi; Geometry; Geometri; algebraic topology}}, language = {{eng}}, publisher = {{Centre for Mathematical Sciences, Lund University}}, school = {{Lund University}}, title = {{Multivariable Orthogonal Polynomials as Coupling Coefficients for Lie and Quantum Algebra Representations}}, year = {{1999}}, }