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25
A Recursion and a Combinatorial Formula for Jack Polynomials
 Invent. Math
, 1997
"... this paper is to add to the existing characterizations of Jack polynomials two further ones: c) a recursion formula among the F together with two formulas to obtain J from them. d) combinatorial formulas of both J and F in terms of certain generalized tableaux ..."
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Cited by 80 (5 self)
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this paper is to add to the existing characterizations of Jack polynomials two further ones: c) a recursion formula among the F together with two formulas to obtain J from them. d) combinatorial formulas of both J and F in terms of certain generalized tableaux
Intertwining Operators and Polynomials Associated With the Symmetric Group
, 1998
"... There is an algebra of commutative differentialdifference operators which is very useful in studying analytic structures invariant under permutation of coordinates. This algebra is generated by the Dunkl operators T i := 1\Gamma(ij) x i \Gammax j , (i = 1; : : : ; N , where (ij) denotes the ..."
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Cited by 22 (5 self)
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There is an algebra of commutative differentialdifference operators which is very useful in studying analytic structures invariant under permutation of coordinates. This algebra is generated by the Dunkl operators T i := 1\Gamma(ij) x i \Gammax j , (i = 1; : : : ; N , where (ij) denotes the transposition of the variables x i ; x j and k is a fixed parameter). We introduce a family of functions fp ff g, indexed by mtuples of nonnegative integers ff = (ff 1 ; : : : ; am ) for m N , which allow a workable treatment of important constructions such as the intertwining operator V . This is a linear map on polynomials, preserving the degree of homogeneity, for which T i V = V ; i = 1; : : : ; N , normalized by V 1 = 1 (see Dunkl, Canadian J. Math. 43(1991), 12131227). We show that T i p ff = 0 for i ? m, and 1 ! 2 ! \Delta \Delta \Delta m ! A fiff p fi ; where ( 1 ; 2 ; : : : ; m ) is the partition whose parts are the entries of ff (That is, 1 2 m 0), fi = (fi 1 ; : : : ; fi m ); i=1 ff i and the sorting of fi is a partition strictly larger than in the dominance order. This triangular matrix representation of V allows a detailed study. There is an inner product structure on spanfp ff g and a convenient set of selfadjoint operators, namely T i ae i , where ae i p ff := p (ff 1 ;:::;ff i +1;:::;ff m) . This structure has a biorthogonal relationship with the Jack polynomials in m variables. Values of k for which V fails to exist are called singular values and were studied by de Jeu, Opdam, and the author in Trans. Amer. Math. Soc. 346(1994), 237256. As a partial verification of a conjecture made in that paper, we construct, for any a = 1; 2; 3; : : : such that gcd(N \Gamma m + 1; a) ! (N \Gamma m + 1)=m and m N=2, a space of polynomials ...
Rodrigues formulas for the Macdonald polynomials
, 1997
"... We present formulas of Rodrigues type giving the Macdonald polynomials for arbitrary partitions through the repeated application of creation operators B k , k = 1; : : : ; `() on the constant 1. Three expressions for the creation operators are derived one from the other. When the last of these e ..."
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Cited by 20 (1 self)
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We present formulas of Rodrigues type giving the Macdonald polynomials for arbitrary partitions through the repeated application of creation operators B k , k = 1; : : : ; `() on the constant 1. Three expressions for the creation operators are derived one from the other. When the last of these expressions is used, the associated Rodrigues formula readily implies the integrality of the (q; t)Kostka coefficients. The proofs given in this paper rely on the connection between affine Hecke algebras and Macdonald polynomials.
qDifference raising operators for Macdonald polynomials and the integrality of transition coefficients
 in Algebraic Methods and qspecial functions
, 1996
"... this paper is study certain qdifference raising operators for Macdonald polynomials (of type A n\Gamma1 ) which are originated from the qdifference  reflection operators introduced in our previous paper [KN]. These operators can be regarded as a qdifference version of the raising operators for J ..."
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Cited by 18 (3 self)
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this paper is study certain qdifference raising operators for Macdonald polynomials (of type A n\Gamma1 ) which are originated from the qdifference  reflection operators introduced in our previous paper [KN]. These operators can be regarded as a qdifference version of the raising operators for Jack polynomials introduced by L. Lapointe and L. Vinet [LV1, LV2]. As an application of our qdifference raising operators, we will give a proof of the integrality of the double Kostka coefficients which had been conjectured by I.G. Macdonald [Ma], Chapter VI. We will also determine their quasiclassical limits, which give rise to (differential) raising operators for Jack polynomials. (See also Notes at the end of
A geometric parametrization for the virtual Euler characteristics of the moduli spaces of real and complex algebraic curves
 Trans. Amer. Math. Soc
"... Abstract. We determine an expression ξs g(γ) for the virtual Euler characteristics of the moduli spaces of spointed real (γ = 1/2) and complex (γ = 1) algebraic curves. In particular, for the space of real curves of genus g with a fixed point free involution, we find that the Euler characteristic i ..."
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Cited by 12 (0 self)
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Abstract. We determine an expression ξs g(γ) for the virtual Euler characteristics of the moduli spaces of spointed real (γ = 1/2) and complex (γ = 1) algebraic curves. In particular, for the space of real curves of genus g with a fixed point free involution, we find that the Euler characteristic is (−2) s−1 (1−2g−1)(g+s−2)!Bg/g! whereBgis the gth Bernoulli number. This complements the result of Harer and Zagier that the Euler characteristic of the moduli space of complex algebraic curves is (−1) s (g+s−2)!Bg+1/(g+1)(g−1)! The proof uses Strebel differentials to triangulate the moduli spaces and some recent techniques for map enumeration to count cells. The approach involves a parameter γ that permits specialization of the formula to the real and complex cases. This suggests that ξs g (γ) itself may describe the Euler characteristics of some related moduli spaces, although we do not yet know what these spaces might be. 1.
The history of qcalculus and a new method
, 2000
"... 1.1. Partitions, generalized Vandermonde determinants and representation theory. 5 1.2. The Frobenius character formulae. 8 ..."
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Cited by 10 (8 self)
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1.1. Partitions, generalized Vandermonde determinants and representation theory. 5 1.2. The Frobenius character formulae. 8
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.
Creation operators for the Macdonald and Jack polynomials
, 1996
"... Formulas of Rodriguestype for the Macdonald polynomials are presented. They involve creation operators, certain properties of which are proved and other conjectured. ..."
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Cited by 6 (1 self)
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Formulas of Rodriguestype for the Macdonald polynomials are presented. They involve creation operators, certain properties of which are proved and other conjectured.
Orthogonal polynomials of types A and B and related Calogero models
 Commun. Math. Phys
, 1998
"... There are examples of CalogeroSutherland models associated to the Weyl groups of type A and B. When exchange terms are added to the Hamiltonians the systems have nonsymmetric eigenfunctions, which can be expressed as products of the ground state with members of a family of orthogonal polynomials. T ..."
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Cited by 5 (1 self)
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There are examples of CalogeroSutherland models associated to the Weyl groups of type A and B. When exchange terms are added to the Hamiltonians the systems have nonsymmetric eigenfunctions, which can be expressed as products of the ground state with members of a family of orthogonal polynomials. These polynomials can be defined and studied by using the differentialdifference operators introduced by the author in Trans. Amer. Math. Soc. 1989 (311), 167183. After a description of known results, particularly from the works of Baker and Forrester, and Sahi; there is a study of polynomials which are invariant or alternating for parabolic subgroups of the symmetric group. The detailed analysis depends on using two bases of polynomials, one of which transforms monomially under group actions and the other one is orthogonal. There are formulas for norms and pointevaluations which are simplifications of those of Sahi. For any parabolic subgroup of the symmetric group there is a skew operator on polynomials which leads to evaluation at (1,1,...,1) of the quotient of the unique skew polynomial in a given irreducible subspace by the minimum alternating polynomial, analogously to a Weyl character formula. The last section concerns orthogonal polynomials for the type B Weyl group with an emphasis on the Hermitetype polynomials. These can be expressed by using the generalized binomial coefficients. A complete basis of eigenfunctions of Yamamoto’s BN spin Calogero model is obtained by multiplying these polynomials by the ground state. 1
Sequences of Symmetric Polynomials and Combinatorial Properties of Tableaux
 Mathematik, RWTH Aachen, D52056
, 1995
"... In 1977 G.P. Thomas has shown that the sequence of Schur polynomials associated to a partition can be comfortabely generated from the sequence of variables x = (x1 ; x2 ; x3 ; : : : ) by the application of a mixed shift/multiplication operator, which in turn can be easily computed from the set S ..."
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Cited by 4 (4 self)
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In 1977 G.P. Thomas has shown that the sequence of Schur polynomials associated to a partition can be comfortabely generated from the sequence of variables x = (x1 ; x2 ; x3 ; : : : ) by the application of a mixed shift/multiplication operator, which in turn can be easily computed from the set SY T () of standard Young tableaux of shape .