T. H. Baker and P. J. Forrester, Nonsymmetric Jack polynomials and integral kernels, Duke Math. J. 95 (1998), 1-50.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....Laguerre polynomials on R . The latter two are related to the CalogeroSutherland models associated to the Weyl groups of type A and type B. 1. Introduction Recently the Calogero Sutherland models associated to the Weyl groups of type A and type B are studied by several authors (cf. [2, 3 ,4, 7, 12, 13, 15, 17] and the references there) The model of type A is the Calogero model with harmonic term, which consists of d particles on the circle with inverse square interaction in an harmonic potential, whose Hamiltonian is, after changing variable x k = exp(i k ) and conjugation with the ground state, ....

....(1.1) HA : 2y j 2 h where (ij) denotes the transposition x i x j . The eigenfunctions of HA are polynomials orthogonal on R with respect to the weight function (y) exp(jyj ) where h (y) where jyj denotes the standard Euclidean norm of y ([3,4]) The weight function h is invariant under the symmetric group S d , a Weyl group of type A. The polynomials are 1991 Mathematics Subject Classi cation. 33C50, 33C45, 33C80. Key words and phrases. Generalized classical orthogonal polynomials, several variables, unit ball, simplex, ....

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T. H. Baker and P. J. Forrester, Nonsymmetric Jack polynomials and integral kernels, Duke Math. J. 95 (1998), 1-50.

....can be studied using the h harmonics; see [48] Apart from some two dimensional examples ( 15] classical and product type orthogonal polynomials are the only cases for which explicit formulae are available. The Hermite type polynomials of type A and type B are studied by Baker and Forrester [4], Lassalle [19] Dunkl [9] and several other people. The commuting self adjoint operators that are used to de ne the nonsymmetric Jack polynomials are due to Cherednik, they are related to the Dunkl operators. The nonsymmetric Jack polynomials are de ned by Opdam [25] There are many other papers ....

T, H. Baker and P. I. Forrester, Nonsymmetric Jack polynomials and integral kernels, Duke Math. J. 95 (1998), 1-50.

....in the expression of the eigenbasis. The eigenbasis of the Calogero Sutherland model is written in terms of the non symmetric Jack polynomials, and the one in the Calogero model is written in terms of the non symmetric Hermite polynomials. These two families of polynomials are deeply connected [3, 4], in particular see the relation (4.21) in this paper. This is the key point why we can treat the Calogero model in a way similar to the one applied to the Calogero Sutherland model in [21] Acknowledgment The author would like to thank D. Uglov for useful comments and support. Thanks are also ....

....; nN ; n 1 1) 4 Creation operators and the non symmetric generalized Hermite polynomials. We will introduce the creation (annihilation) operators, the Dunkl operators for the Calogero model, and the non symmetric generalized Hermite polynomials. These polynomials are also introduced in [3, 4] in a little different way. In this section, we will deal with the following scalar product (2.9) hf; gi c : N Y i=1 Z 1 01 dx i Y j k jx j 0 x k j 2 f(x 1 ; x 2 ; xN )g(x 1 ; x 2 ; xN ) First we can check that d y i = 0d i ; x y i = x i : 4.1) We define the ....

T. H. Baker, and P. J. Forrester,: Non-Symmetric Jack Polynomials and Integral Kernels, Preprint (

....is intimately connected with the corresponding algebra of Dunkl operators. Since then, there has been an extensive and ongoing study of CMS models and explicit operator solutions for them via di erentialre ection operator formalisms; among the broad literature, we refer to [L V] K] BHKV] [BF], and [U W] Let us brie y describe the connection of abstract Calogero models with Dunkl operators: Consider the following modi cation of e F k , involving re ection terms: F k = 2 X 2R k( h ; xi 2 (k( 3.2) In order to avoid singularities in the re ecting hyperplanes, it ....

Baker, T.H., Forrester, P.J., Non-symmetric Jack polynomials and integral kernels. Duke Math. J. 95 (1998), 1-50.

.... During the last years, these operators have gained considerable interest in various fields of mathematics and also in physical applications; they are, for example, naturally connected with certain Schrodinger operators for Calogero Sutherlandtype quantum many body systems, see [L V] and [B F2] [B F3]. For a finite reflection group G ae O(N;R) on R N the associated Dunkl operators are defined as follows: For ff 2 R N n f0g , denote by oe ff the reflection corresponding to ff , i.e. in the hyperplane This paper was written while the author held a Forschungsstipendium of the DFG at the ....

.... G = Z 2 on R , our generalized Hermite polynomials coincide with those introduced in [Chi] and studied in [Ros] Our setting also includes, for the symmetric group G = SN , the so called non symmetric generalized Hermite polynomials which were recently introduced by Baker and Forrester in [B F2] [B F3]. These are non symmetric analogues of the symmetric, i.e. permutation invariant generalized Hermite polynomials associated with the group SN , which were first introduced by Lassalle in [L2] Moreover, the generalized Laguerre polynomials of [B F2] B F3] which are non symmetric analogues of ....

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Baker, T.H., Forrester, P.J.: Non-symmetric Jack polynomials and integral kernels. Duke J. Math., to appear.

....derivatives, which are associated with a finite reflection group on some Euclidean space. They play, for example, a useful role in the algebraic description of exactly solvable quantum many body systems of Calogero MoserSutherland type; among the broad literature in this context, we refer to [B F], L V] and [P] In his paper [dJ2] de Jeu proved a quite general uncertainty principle for integral operators with bounded kernel which applies to the Dunkl transform; this result has the form of an ffl Gamma ffi concentration principle as first stated in [D S] for the Fourier transform. ....

T.H. Baker and P.J. Forrester, Non-symmetric Jack polynomials and integral kernels. Duke Math. J., to appear.

.... of degenerate affine Hecke algebras (see [C] O2] and, for some background, Ki] Moreover, they have been succesfully involved in the description and solution of Calogero Moser Sutherland type quantum many body systems; among the wide literature in this context, we refer to [P] L V] and [B F]. Let G ae O(N;R) be a finite reflection group on R N . For ff 2 R N n f0g , we denote by oe ff the reflection in the hyperplane orthogonal to ff , i.e. oe ff (x) x Gamma 2 hff; xi jffj 2 ff ; where h: i denotes the Euclidean scalar product on R N and jxj : p hx; xi . We use the ....

Baker, T.H. & Forrester, P.J., Non-symmetric Jack polynomials and integral kernels. Duke Math. J., to appear.

....Mcdonald and Mehta (see [Me] and replace the classical Gaussian in the Dunkl theory. We mention that systems of orthogonal polynomials related to Dunkl s Gaussian distributions on R N , called generalized Hermite polynomials, have been studied in various contexts during the last years, see e. g [B F, vD, R2, Ros] and references therein. In the case of the symmetric group, the generalized Hermite polynomials play a role in quantum many body systems of Calogero Moser Sutherland type and are closely related to Jack polynomials. The biorthogonal systems of this paper are introduced in a quite canonical way; ....

.... forms a system of orthogonal polynomials with respect to P Gamma t (0; These generalized Hermite polynomials are studied (for t = 1=4) in [R2] and their relations to the Appell systems (R ) 2Z N and (S ) 2Z N are studied in [R V] We also refer to related investigations in [B F, vD] and references given there. ....

T.H. Baker, P.J. Forrester, Non-symmetric Jack polynomials and integral kernels, Duke Math. J., to appear.

....(2.6) and s i E j = E j (1 Gamma ffi i;j )E s i j j i j i 1 E j j i = j i 1 E j E s i j j i j i 1 (2.7) where ffi i;j : j i Gamma j i 1 . Using the operators Phi and s i , it is very simple to establish by recurrence formulas for E j (1 ) 11] and hE j ; E j i I [3]. To write down these formulas requires some notation. Following Sahi [11] for a node s = i; j) in the diagram of a composition define the arm length a(s) arm colength a (s) leg length l(s) and leg colength l (s) by a(s) j i Gamma j l(s) #fk ijj j k j i g #fk ijj j k 1 ....

....kN=ff. Also, expanding the l.h.s. as a power series we see that each term is a polynomial in kN=ff. Since both sides are equal for each k 2 ZZ , they must in fact be equal for all (complex) values of kN=ff = r Gamma 1. Thus (2.13) can be rewritten as This is eq. 2.13) of ref. [3] with d j =f j 7 u j =d j : 2.15) Now independent of the particular value of u j the reasoning of ref. 3] which relies on (simple) properties of the E j , remains valid and gives the integration formula (2.19) of ref. 3] with the replacement (2.15) the asymptotics of which implies that jjj ....

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T. H. Baker and P. J. Forrester, "Non--symmetric Jack polynomials and integral kernels", qalg /9612003, to appear in Duke J. Math.

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T. H. Baker and P. J. Forrester. Non--symmetric Jack polynomials and integral kernels. q-alg/9612003, to appear in Duke J. Math.

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Baker, T.H., Forrester, P.J., Non-symmetric Jack polynomials and integral kernels. Duke Math. J. 95 (1998), 150.

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Baker T.H., Forrester, P.J.: Non-symmetric Jack polynomials and integral kernels. Duke J. Math., to appear.

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