Orthogonal Polynomials Associated with Root Systems

MacDonald, Ian G.

doi:10.1007/978-94-009-0501-6_14

Cited by 178 publications

(212 citation statements)

References 3 publications

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“…Open questions abound for the type II and IV systems, too. Even in the type III case, where transforms in terms of multivariable orthogonal polynomials are known both for the commuting PDOs [14,15] and for the commuting A.1.0s [16,17], the polynomials are not known in a sufficiently explicit way to establish the duality properties expected from the classical level (save for N = 2 [5]). …”

Section: 42)-(546)mentioning

confidence: 99%

“…The intertwining relations 17) show that Ui 1 can indeed be viewed as a linearizing canonical transformation <1>15: n--+ !1°; cf. also (2.11), (2.12).…”

Section: Wave Maps and Pure Soliton Systemsmentioning

confidence: 99%

“…Macdonald [16,17]. To be more specific, he considers several root systems at once and accordingly works ,with the center-of-mass versions of the Bk and Qm; for the detailed relation we refer to Section 5.2 of Ref.…”

Section: First It Satisfiesmentioning

confidence: 99%

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Systems of Calogero-Moser Type

Ruijsenaars¹

1999

Particles and Fields

161

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We survey results on Galilei-and Poincare-invariant CalogeroMoser and Toda N-particle systems, both in the context of classical mc hanics and of quantum mechanics. Special attention is given to integrability issues and interconnections between the various models. Action-angle and joint eigenfunction transforms are also considered, and some novel results on N = 2 eigenfunctions of hyperbolic Askey-Wilson type and of relativistic elliptic type are sketched.

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Section: 42)-(546)mentioning

confidence: 99%

“…The intertwining relations 17) show that Ui 1 can indeed be viewed as a linearizing canonical transformation <1>15: n--+ !1°; cf. also (2.11), (2.12).…”

Section: Wave Maps and Pure Soliton Systemsmentioning

confidence: 99%

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Systems of Calogero-Moser Type

Ruijsenaars¹

1999

Particles and Fields

161

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“…Macdonald polynomials were first introduced by I. G. Macdonald in the late 1980s (see [14]) and continue to make important appearances in a variety of fields such as algebraic geometry, physics, representation theory, and combinatorics. They provide an example of a family of symmetric functions, that is, they are invariant under all permutations of the variables.…”

Section: Branching Rules For Classical Groups and Generalizationsmentioning

confidence: 99%

Vanishing integrals for Hall–Littlewood polynomials

Venkateswaran

2012

Transformation Groups

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It is well known that if one integrates a Schur function indexed by a partition λ over the symplectic (resp. orthogonal) group, the integral vanishes unless all parts of λ have even multiplicity (resp. all parts of λ are even). In a recent paper of Rains and Vazirani, Macdonald polynomial generalizations of these identities and several others were developed and proved using Hecke algebra techniques. However at q = 0 (the Hall-Littlewood level), these approaches do not work, although one can obtain the results by taking the appropriate limit. In this paper, we develop a direct approach for dealing with this special case. This technique allows us to prove some identities that were not amenable to the Hecke algebra approach. Moreover, we are able to generalize some of the identities by introducing extra parameters. This leads us to a finite-dimensional analog of a recent result of Warnaar, which uses the Rogers-Szegő polynomials to unify some existing summation type formulas for Hall-Littlewood functions.

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“…Being a modification of monomial symmetric functions, they are directly related to the theory of symmetric (Laurent) polynomials [14,15,16,17] (see Section 11 in [7]). …”

Section: Introductionmentioning

confidence: 99%

E-Orbit Functions

Klimyk¹,

Patera

2008

SIGMA

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Abstract. We review and further develop the theory of E-orbit functions. They are functions on the Euclidean space E n obtained from the multivariate exponential function by symmetrization by means of an even part W e of a Weyl group W , corresponding to a CoxeterDynkin diagram. Properties of such functions are described. They are closely related to symmetric and antisymmetric orbit functions which are received from exponential functions by symmetrization and antisymmetrization procedure by means of a Weyl group W . The E-orbit functions, determined by integral parameters, are invariant with respect to even part W aff e of the affine Weyl group corresponding to W . The E-orbit functions determine a symmetrized Fourier transform, where these functions serve as a kernel of the transform. They also determine a transform on a finite set of points of the fundamental domain F e of the group W aff e (the discrete E-orbit function transform).

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Orthogonal Polynomials Associated with Root Systems

Cited by 178 publications

References 3 publications

Systems of Calogero-Moser Type

Systems of Calogero-Moser Type

Vanishing integrals for Hall–Littlewood polynomials

E-Orbit Functions

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