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86
On Inner Product In Modular Tensor Categories. II Inner Product On Conformal Blocks.
 I & II, math.QA/9508017 and qalg/9611008
, 1995
"... this paper, we apply the same construction to the MTC coming from the integrable representations of affine Lie algebras or, equivalently, from WessZuminoWitten model of conformal field theory. We briefly recall construction of this category, first suggested by Moore and Seiberg (see [MS1,2]) and la ..."
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this paper, we apply the same construction to the MTC coming from the integrable representations of affine Lie algebras or, equivalently, from WessZuminoWitten model of conformal field theory. We briefly recall construction of this category, first suggested by Moore and Seiberg (see [MS1,2]) and later refined by Kazhdan and Lusztig ([KL14]) and Finkelberg ([F]) in Section 9. In particular, spaces of homomorphisms in this category are the spaces of conformal blocks of WZW model. Thus, the general theory developed in Section 2 of [K] gives us an inner product on the space of conformal blocks, and so defined inner product is modular invariant. This definition is constructive: we show how it can be rewritten so that it only involves Drinfeld associator, or, equivalently, asymptotics of solutions of KnizhnikZamolodchikov equations. Since there are integral formulas for the solutions of KZ equations, this shows that the inner product on the space of conformal blocks can be written explicitly in terms of certain integrals. In the case g = sl 2 these integrals can be calculated (see [V]), using Selberg integral, and the answer is written in terms of \Gammafunctions. Thus, in this case we can write explicit formulas for inner product on the space of conformal blocks. These expressions are 1991 Mathematics Subject Classification. Primary 81R50, 05E35, 18D10; Secondary 57M99
Macdonald's Polynomials And Representations Of Quantum Groups
 Math. Res. Let
, 1994
"... this paper we present a formula for Macdonald's polynomials which arises from the representation theory of the quantum group U q (gl n ). This formula expresses Macdonald's polynomials as vectorvalued characters  (weighted) traces of intertwining operators between certain modules over U ..."
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Cited by 45 (12 self)
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this paper we present a formula for Macdonald's polynomials which arises from the representation theory of the quantum group U q (gl n ). This formula expresses Macdonald's polynomials as vectorvalued characters  (weighted) traces of intertwining operators between certain modules over U q (gl n ). This result was announced in [EK]. It is an interesting problem to find relation between this construction and a recent paper of Noumi ([No]) which gives interpretation of Macdonald's polynomials for special values of k as zonal spherical functions on a homogeneous space for a quantum group
Macdonald's Evaluation Conjectures and Difference Fourier Transform
, 1994
"... This paper contains the proof of the remaining two (the duality and evaluation conjectures). The evaluation conjecture (now a theorem) is in fact a q; tgeneralization of the classic Weyl dimension formula. One can expect interesting applications of this theorem since the socalled qdimensions are ..."
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Cited by 44 (2 self)
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This paper contains the proof of the remaining two (the duality and evaluation conjectures). The evaluation conjecture (now a theorem) is in fact a q; tgeneralization of the classic Weyl dimension formula. One can expect interesting applications of this theorem since the socalled qdimensions are undoubtedly important. It is likely that we can incorporate the KacMoody case as well. The necessary technique was developed in [C4]. As to the duality theorem (in its complete form), it states that the generalized trigonometricdifference zonal Fourier transform is selfdual (at least formally). We define this q; ttransform in terms of double affine Hecke algebras. The most natural way to check the selfduality is to use the connection of these algebras with the socalled elliptic braid groups (the Fourier involution will turn into the transposition of the periods of an elliptic curve). The classical trigonometricdifferential Fourier transform (corresponding to the limit q = t as t ! 1 for certain special k) plays one of the main roles in the harmonic analysis on symmetric spaces. It sends symmetric trigonometric polynomials to the corresponding radial parts of Laplace operators (HarishChandra, Helgason) and is not selfdual. The calculation of its inverse (the Plancherel theorem) is always challenging and involving. * Partially supported by NSF grant DMS9301114 In the rationaldifferential setting, Charles Dunkl introduced the generalized Hankel transform which appeared to be selfdual [D,J]. We demonstrate in this paper that one can save this very important property if trigonometric polynomials come together with difference operators. At the moment, it is mostly an algebraic observation (the differenceanalitical aspects were not touched upon). The root systems of ...
Integrality of Two Variable Kostka Functions
 J. Reine Angew. Math
, 1997
"... this paper is 5.4. Theorem. For m l() consider the expansion E = c ~ m . Then the coefficients c are in Z[q; t] ..."
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this paper is 5.4. Theorem. For m l() consider the expansion E = c ~ m . Then the coefficients c are in Z[q; t]
Inversion of the Pieri formula for Macdonald polynomials
 Adv. Math
"... We give the explicit analytic development of Macdonald polynomials in terms of “modified complete ” and elementary symmetric functions. These expansions are obtained by inverting the Pieri formula. Specialization yields similar developments for monomial, Jack and Hall–Littlewood symmetric functions. ..."
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We give the explicit analytic development of Macdonald polynomials in terms of “modified complete ” and elementary symmetric functions. These expansions are obtained by inverting the Pieri formula. Specialization yields similar developments for monomial, Jack and Hall–Littlewood symmetric functions.
Nonsymmetric Macdonald's polynomials
, 1995
"... ed to the same questions in the symmetric case. The main point is the definition of the difference spherical Fourier transform based on the double affine Hecke algebras (generalizing the Hankel transform from [D,J]). For instance, it readily results in the normformulas conjectured by Macdonald and p ..."
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ed to the same questions in the symmetric case. The main point is the definition of the difference spherical Fourier transform based on the double affine Hecke algebras (generalizing the Hankel transform from [D,J]). For instance, it readily results in the normformulas conjectured by Macdonald and proved in [C2] and clarifies why they are so surprisingly simple. Hopefully, this version of the Macdonald theory can be extended to the elliptic case (see [C6]) without serious difficulties. * Partially supported by NSF grant DMS9301114 Once the Fourier transform appeared, we cannot restrict ourselves to symmetric functions anymore. Even the classical multicomponent Fourier transform requires at least the coordinate functions and the corresponding differentiations. It reveals itself at many levels. First, it is easier to operate with the double affine Hecke algebra than with its (very complicated) subalgebra of symmetric operators. Second, promising applications are expected in arithme
Three formulas for eigenfunctions of integrable Schrödinger operators
, 1995
"... Abstract. We give three formulas for meromorphic eigenfunctions (scattering states) of Sutherland’s integrable Nbody Schrödinger operators and their generalizations. The first is an explicit computation of the Etingof–Kirillov traces of intertwining operators, the second an integral representation ..."
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Abstract. We give three formulas for meromorphic eigenfunctions (scattering states) of Sutherland’s integrable Nbody Schrödinger operators and their generalizations. The first is an explicit computation of the Etingof–Kirillov traces of intertwining operators, the second an integral representation of hypergeometric type, and the third of Bethe ansatz type. The last two formulas are degenerations of elliptic formulas obtained previously in connection with the Knizhnik–Zamolodchikov– Bernard equation. The Bethe ansatz formulas in the elliptic case are reviewed and discussed in more detail here: Eigenfunctions are parametrized by a “Hermite–Bethe ” variety, a generalization of the spectral variety of the Lamé operator. We also give the qdeformed version of our first formula. In the scalar slN case, this gives common eigenfunctions of the commuting Macdonald–Rujsenaars difference operators. 1.
TRACES OF INTERTWINERS FOR QUANTUM GROUPS AND DIFFERENCE EQUATIONS, I
 VOL. 104, NO. 3 DUKE MATHEMATICAL JOURNAL
, 2000
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A Remarkable q,tCatalan Sequence and qLagrange Inversion
"... We introduce a rational function C n (q; t) and conjecture that it always evaluates to a polynomial in q; t with nonnegative integer coefficients summing to the familiar Catalan number \Gamma 2n \Delta . We give supporting evidence by computing the specializations D n (q) = C n (q; 1=q) ..."
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Cited by 28 (8 self)
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We introduce a rational function C n (q; t) and conjecture that it always evaluates to a polynomial in q; t with nonnegative integer coefficients summing to the familiar Catalan number \Gamma 2n \Delta . We give supporting evidence by computing the specializations D n (q) = C n (q; 1=q) q and C n (q) = C n (q; 1) = C n (1; q). We show that, in fact, D n (q) qcounts Dyck words by the major index and C n (q) qcounts Dyck paths by area. We also show that C n (q; t) is the coefficient of the elementary symmetric function e n in a symmetric polynomial DH n (x; q; t) which is the conjectured Frobenius characteristic of the module of diagonal harmonic polynomials. On the validity of certain conjectures this yields that C n (q; t) is the Hilbert series of the diagonal harmonic alternants. It develops that the specialization DH n (x; q; 1) yields a novel and combinatorial way of expressing the solution of the qLagrange inversion problem studied by Andrews [2], Garsia [5] and Gessel [11]. Our proofs involve manipulations with the Macdonald basis fP (x; q; t)g which are best dealt with inring notation. In particular we derive here thering version of several symmetric function identities.