The Jack polynomials with prescribed symmetry are obtained from the nonsymmetric polynomials via the operations of symmetrization, antisymmetrization and normalization. After dividing out the corresponding antisymmetric polynomial of smallest degree, a symmetric polynomial results. Of interest in applications is the value of the latter polynomial when all the variables are set equal. Dunkl has obtained this evaluation, making use of a certain skew symmetric operator. We introduce a simpler operator for this purpose, thereby obtaining a new derivation of the evaluation formula. An expansion formula of a certain product in terms of Jack polynomials with prescribed symmetry implied by the evaluation formula is used to derive a generalization of a constant term identity due to Macdonald, Kadell and Kaneko. Although we don't give the details in this work, the operator introduced here can be defined for any reduced crystallographic root system, and used to provide an evaluation formula for the corresponding Heckman-Opdam polynomials with prescribed symmetry.
We study the hole propagator of the Calogero-Sutherland model with SU(2) internal symmetry. We obtain the exact expression for arbitrary non-negative integer coupling parameter $β$ and prove the conjecture proposed by one of the authors. Our method is based on the theory of the Jack polynomials with a prescribed symmetry.
The Macdonald polynomials with prescribed symmetry are obtained from the nonsymmetric Macdonald polynomials via the operations of $t$-symmetrisation, $t$-antisymmetrisation and normalisation. Motivated by corresponding results in Jack polynomial theory we proceed to derive an expansion formula and a related normalisation. Eigenoperator methods are used to relate the symmetric and antisymmetric Macdonald polynomials, and we discuss how these methods can be extended to special classes of the prescribed symmetry polynomials in terms of their symmetric counterpart. We compute the explicit form of …
In the theory of symmetric Jack polynomials the coefficients in the expansion of the $p$th elementary symmetric function $e_p(z)$ times a Jack polynomial expressed as a series in Jack polynomials are known explicitly. Here analogues of this result for the non-symmetric Jack polynomials $E_η(z)$ are explored. Necessary conditions for non-zero coefficients in the expansion of $e_p(z) E_η(z)$ as a series in non-symmetric Jack polynomials are given. A known expansion formula for $z_i E_η(z)$ is rederived by an induction procedure, and this expansion is used to deduce the corresponding result for t…
Affine analogue of Jack's polynomials introduced by Etingof and Kirillov Jr. is studied for the case of \hat{sl}_2. Using the Wakimoto representation, we give an integral formula of elliptic Selberg type for the affine Jack's polynomials. From this integral formula, the action of SL_2(Z) on the space of the affine Jack's polynomials is computed. For simple cases, we write down the affine Jack's polynomials in terms of some modular and elliptic functions.
We use a new method to study the Laplace-Beltrami type operator on the Fock space of symmetric functions, and as an example of our explicit computation we show that the Jack symmetric functions are the only family of eigenvectors of the differential operator. As applications of this explicit method we find a combinatorial formula for Jack symmetric functions and the Littlewood-Richardson coefficients in the Jack case. As further applications, we obtain a new determinantal formula for Jack symmetric functions. We also obtained a generalized raising operator formula for Jack symmetric functions,…
Formulas of Rodrigues-type for the Macdonald polynomials are presented. They involve creation operators, certain properties of which are proved and other conjectured. The limiting case of the Jack polynomials is discussed.
The Hi-Jack symmetric polynomials, which are associated with the simultaneous eigenstates for the first and second conserved operators of the quantum Calogero model, are studied. Using the algebraic properties of the Dunkl operators for the model, we derive the Rodrigues formula for the Hi-Jack symmetric polynomials. Some properties of the Hi-Jack polynomials and the relationships with the Jack symmetric polynomials and with the basis given by the QISM approach are presented. The Hi-Jack symmetric polynomials are strong candidates for the orthogonal basis of the quantum Calogero model.
We give an iterative method to realize general Jack functions from Jack functions of rectangular shapes. We first show some cases of Stanley's conjecture on positivity of the Littlewood-Richardson coefficients, and then use this method to give a new realization of Jack functions. We also show in general that vectors of products of Jack vertex operators form a basis of symmetric functions. In particular this gives a new proof of linear independence for the rectangular and marked rectangular Jack vertex operators. Thirdly a generalized Frobenius formula for Jack functions was given and was used …
We give explicit formulae of Whittaker vectors for Virasoro algebra in terms of the Jack symmetric polynomials. Our fundamental tools are the Feigin-Fuchs bosonization and the split expression of the Calogero-Sutherland model given by Awata-Matsuo-Odake-Shiraishi.
On the vertex operator algebra associated with rank one lattice we derive a general formula for products of vertex operators in terms of generalized homogeneous symmetric functions. As an application we realize Jack symmetric functions of rectangular shapes as well as marked rectangular shapes.
We call superpartitions the indices of the eigenfunctions of the supersymmetric extension of the trigonometric Calogero-Moser-Sutherland model. We obtain an ordering on superpartitions from the explicit action of the model's Hamiltonian on monomial superfunctions. This allows to define Jack superpolynomials as the unique eigenfunctions of the model that decompose triangularly, with respect to this ordering, on the basis of monomial superfunctions. This further leads to a simple and explicit determinantal expression for the Jack superpolynomials.
Symmetric Jack polynomials arise naturally in several contexts, including statistics, physics, combinatorics, and representation theory. They are pairwise orthogonal with repsect to two different inner products, the first defined by integration over an n-dimensional torus, and the second defined via a power series expansion. The nonsymmetric analogs of Jack polynomials were recently defined by Opdam via the first inner product. In this paper we show that they are also orthogonal with respect to an analog of the second product which was proposed by Dunkl. We also derive an explict formula for t…
We consider the coincident root loci consisting of the polynomials with at least two double roots andpresent a linear basis of the corresponding ideal in the algebra of symmetric polynomials in terms of the Jack polynomials with special value of parameter $α= -2.$ As a corollary we present an explicit formula for the Hilbert-Poincarè series of this ideal and the generator of the minimal degree as a special Jack polynomial. A generalization to the case of the symmetric polynomials vanishing on the double shifted diagonals and the Macdonald polynomials specialized at $t^2 q = 1$ is also presente…
It is shown that the deformed Calogero-Moser-Sutherland (CMS) operators can be described as the restrictions on certain affine subvarieties (called generalised discriminants) of the usual CMS operators for infinite number of particles. The ideals of these varieties are shown to be generated by the Jack symmetric functions related to the Young diagrams with special geometry. A general structure of the ideals which are invariant under the action of the quantum CMS integrals is discussed in this context. The shifted super-Jack polynomials are introduced and combinatorial formulas for them and for…
A new generalization of the Jack polynomials that incorporates fermionic variables is presented. These Jack superpolynomials are constructed as those eigenfunctions of the supersymmetric extension of the trigonometric Calogero-Moser-Sutherland (CMS) model that decomposes triangularly in terms of the symmetric monomial superfunctions. Many explicit examples are displayed. Furthermore, various new results have been obtained for the supersymmetric version of the CMS models: the Lax formulation, the construction of the Dunkl operators and the explicit expressions for the conserved charges. The ref…
The wave functions of the Calogero-Sutherland model are known to be expressible in terms of Jack polynomials. A formula which allows to obtain the wave functions of the excited states by acting with a string of creation operators on the wave function of the ground state is presented and derived. The creation operators that enter in this formula of Rodrigues-type for the Jack polynomials involve Dunkl operators.
We give the explicit analytic development of any Jack or Macdonald polynomial in terms of elementary (resp. modified complete) symmetric functions. These two developments are obtained by inverting the Pieri formula.
The Calogero model is a one-dimensional quantum integrable system with inverse-square long-range interactions confined in an external harmonic well. It shares the same algebraic structure with the Sutherland model, which is also a one-dimensional quantum integrable system with inverse-sine-square interactions. Inspired by the Rodrigues formula for the Jack polynomials, which form the orthogonal basis of the Sutherland model, recently found by Lapointe and Vinet, we construct the Rodrigues formula for the Hi-Jack (hidden-Jack) polynomials that form the orthogonal basis of the Calogero model.
We extend some results about shifted Schur functions to the general context of shifted Macdonald polynomials. We obtain two explicit formulas for these polynomials: a $q$-integral representation and a combinatorial formula. Our main tool is a $q$-integral representation for ordinary Macdonald polynomials. We also discuss duality for shifted Macdonald polynomials and Jack degeneration of these polynomials.
The clustering properties of Jack polynomials are relevant in the theoretical study of the fractional Hall states. In this context, some factorization properties have been conjectured for the $(q,t)$-deformed problem involving Macdonald polynomials. The present paper is devoted to the proof of this formula. To this aim we use four families of Jack/Macdonald polynomials: symmetric homogeneous, nonsymmetric homogeneous, shifted symmetric and shifted nonsymmetric.
