Results 1  10
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19
Connection coefficients, matchings, maps and combinatorial conjectures for Jack symmetric functions
 TRANS. AMER. MATH. SOC
, 1996
"... A power series is introduced that is an extension to three sets of variables of the Cauchy sum for Jack symmetric functions in the Jack parameter α. We conjecture that the coefficients of this series with respect to the power sum basis are nonnegative integer polynomials in b, the Jack parameter sh ..."
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Cited by 11 (6 self)
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A power series is introduced that is an extension to three sets of variables of the Cauchy sum for Jack symmetric functions in the Jack parameter α. We conjecture that the coefficients of this series with respect to the power sum basis are nonnegative integer polynomials in b, the Jack parameter shifted by 1. More strongly, we make the MatchingsJack Conjecture, that the coefficients are counting series in b for matchings with respect to a parameter of nonbipartiteness. Evidence is presented for these conjectures and they are proved for two infinite families. The coefficients of a second series, essentially the logarithm of the first, specialize at values 1 and 2 of the Jack parameter to the numbers of hypermaps in orientable and locally orientable surfaces, respectively. We conjecture that these coefficients are also nonnegative integer polynomials in b, andwemake the HypermapJack Conjecture, that the coefficients are counting series in b for hypermaps in locally orientable surfaces with respect to a parameter of nonorientability.
MOPS: Multivariate Orthogonal Polynomials (symbolically). 2004. Preprint found at lanl.arxiv.org/abs/mathph/0409066
"... In this paper we present a Maple library (MOPs) for computing Jack, Hermite, Laguerre, and Jacobi multivariate polynomials, as well as eigenvalue statistics for the Hermite, Laguerre, and Jacobi ensembles of Random Matrix theory. We also compute multivariate hypergeometric functions, and offer both ..."
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Cited by 10 (6 self)
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In this paper we present a Maple library (MOPs) for computing Jack, Hermite, Laguerre, and Jacobi multivariate polynomials, as well as eigenvalue statistics for the Hermite, Laguerre, and Jacobi ensembles of Random Matrix theory. We also compute multivariate hypergeometric functions, and offer both symbolic and numerical evaluations for all these quantities. We prove that all algorithms are welldefined, analyze their complexity, and illustrate their performance in practice. Finally, we also present a few of the possible applications of this library.
Quantum vs Classical Integrability in RuijsenaarsSchneider Systems
, 2003
"... The relationship (resemblance and/or contrast) between quantum and classical integrability in RuijsenaarsSchneider systems, which are one parameter deformation of CalogeroMoser systems, is addressed. Many remarkable properties of classical Calogero and Sutherland systems (based on any root system) ..."
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Cited by 8 (5 self)
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The relationship (resemblance and/or contrast) between quantum and classical integrability in RuijsenaarsSchneider systems, which are one parameter deformation of CalogeroMoser systems, is addressed. Many remarkable properties of classical Calogero and Sutherland systems (based on any root system) at equilibrium are reported in a previous paper (CorriganSasaki). For example, the minimum energies, frequencies of small oscillations and the eigenvalues of Lax pair matrices at equilibrium are all “integer valued”. In this paper we report that similar features and results hold for the RuijsenaarsSchneider type of integrable systems based on the classical root systems.
Supersymmetric CalogeroMoserSutherland models: superintegrability structure and aigenfunctions, to appear
 in the proceedings of the Workshop on superintegrability in classical and quantum systems, ed. P Winternitz, CRM series
"... 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 CalogeroMoserSutherland (CMS) model that decomposes triangularly in terms of the ..."
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Cited by 7 (6 self)
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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 CalogeroMoserSutherland (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 reformulation of the models in terms of the exchangeoperator formalism is a crucial aspect of our analysis.
The random matrix technique of ghosts and shadows
 Markov Processes and Related Fields
"... We propose to abandon the notion that a random matrix has to be sampled for it to exist. Much of today's applied nite random matrix theory concerns real or complex random matrices (β = 1, 2). The threefold way so named by Dyson in 1962 [2] adds quaternions (β = 4). While it is true there are only th ..."
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Cited by 3 (3 self)
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We propose to abandon the notion that a random matrix has to be sampled for it to exist. Much of today's applied nite random matrix theory concerns real or complex random matrices (β = 1, 2). The threefold way so named by Dyson in 1962 [2] adds quaternions (β = 4). While it is true there are only three real division algebras (β = dimension over the reals), this mathematical fact while critical in some ways, in other ways is irrelevant and perhaps has been over interpreted over the decades. We introduce the notion of a ghost random matrix quantity that exists for every beta, and a shadow quantity which may be real or complex which allows for computation. Any number of computations have successfully given reasonable answers to date though di culties remain in some cases. Though it may seem absurd to have a three and a quarter dimensional or pi dimensional algebra, that is exactly what we propose and what we compute with. In the end β becomes a noisiness parameter rather than a dimension. 1
A2 Macdonald polynomials: a separation of variables; qalg/9512003
"... In this paper we construct a discrete linear operator K which transforms A2 Macdonald polynomials into the product of two basic 3φ2 hypergeometric series with known arguments. The action of the operator K on power sums in two variables can be reduced to a generalization of one particular case of the ..."
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Cited by 1 (0 self)
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In this paper we construct a discrete linear operator K which transforms A2 Macdonald polynomials into the product of two basic 3φ2 hypergeometric series with known arguments. The action of the operator K on power sums in two variables can be reduced to a generalization of one particular case of the Bailey’s summation formula for a verywellpoised 6ψ6 series. We also propose the conjecture for a transformation of 6ψ6 series with different arguments. 1
On Kerov polynomials for Jack characters †
"... Abstract. We consider a deformation of Kerov character polynomials, linked to Jack symmetric functions. It has been introduced recently by M. Lassalle, who formulated several conjectures on these objects, suggesting some underlying combinatorics. We give a partial result in this direction, showing t ..."
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Abstract. We consider a deformation of Kerov character polynomials, linked to Jack symmetric functions. It has been introduced recently by M. Lassalle, who formulated several conjectures on these objects, suggesting some underlying combinatorics. We give a partial result in this direction, showing that some quantities are polynomials in the Jack parameter α with prescribed degree. Our result has several interesting consequences in various directions. Firstly, we give a new proof of the fact that the coefficients of Jack polynomials expanded in the monomial or powersum basis depend polynomially in α. Secondly, we describe asymptotically the shape of random Young diagrams under some deformation of Plancherel measure. Résumé. On considère une déformation des polynômes de Kerov pour les caractères du groupe symétrique. Cette déformation est liée aux polynômes de Jack. Elle a été récemment définie par M. Lassalle, qui a proposé plusieurs conjectures sur ces objets, suggérant ainsi l’existence d’une combinatoire sousjacente. Nous donnons un résultat partiel dans cette direction, en montrant que certaines quantités sont des polynômes (dont on contrôle les degrés) en fonction du paramètre de Jack α. Notre résultat a des conséquences intéressantes dans des directions diverses. Premièrement, nous donnons une nouvelle preuve de la polynomialité (toujours en fonction de α) des coefficients du développement des polynômes de Jack dans la base monomiale. Deuxièmement, nous décrivons asymptotiquement la forme de grands diagrammes de Young distribués selon une déformation de la mesure de Plancherel.
DAMTP/0391 hepth/0309180 Conformal Partial Waves and the Operator Product Expansion
, 2003
"... By solving the two variable differential equations which arise from finding the eigenfunctions for the Casimir operator for O(d, 2) succinct expressions are found for the functions, conformal partial waves, representing the contribution of an operator of arbitrary scale dimension ∆ and spin ℓ togeth ..."
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By solving the two variable differential equations which arise from finding the eigenfunctions for the Casimir operator for O(d, 2) succinct expressions are found for the functions, conformal partial waves, representing the contribution of an operator of arbitrary scale dimension ∆ and spin ℓ together with its descendants to conformal four point functions for d = 4, recovering old results, and also for d = 6. The results are expressed in terms of ordinary hypergeometric functions of variables x, z which are simply related to the usual conformal invariants. An expression for the conformal partial wave amplitude valid for any dimension is also found in terms of a sum over two variable symmetric Jack polynomials which is used to derive relations for the conformal partial waves. PACS no: 11.25.Hf