## Physics and Combinatorics 2000: Proceedings of the Nagoya 2000 International Workshop : Graduate School of Mathematics, Nagoya University, 21-26 August, 2000The Nagoya 2000 International Workshop gathered together a group of scientists actively working in combinatorics, representation theory, special functions, number theory and mathematical physics, to acquaint the participants with some basic results in their fields and to discuss existing and possible interactions between the mentioned subjects. This volume constitutes the proceedings of the workshop. Contents: Vanishing Theorems and Character Formulas for the Hilbert Scheme of Points in the Plane (M Haiman); Exclusion Statistics and Chiral Partition Function (K Hikami); On the Spectrum of Dehn Twists in Quantum Teichmller Theory (R Kashaev); Introduction to Tropical Combinatorics (A Kirillov); Transition on Grothendieck Polynomials (A Lascoux); Generalized HAlder''s Theorem for Multiple Gamma Function (M Nishizawa); Quantum Calogero-Moser Models: Complete Integrability for All the Root Systems (R Sasaki); Simplification of Thermodynamic BetheOCoAnsatz Equations (M Takahashi); and other papers. Readership: Researchers and graduates in mathematical physics and combinatorics & graph theory." |

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### Contents

22 | |

Forresters Constant Term Conjecture and its analogue | 49 |

On the Spectrum of Dehn Twists in Quantum Teichmtiller Theory | 63 |

Introduction to Tropical Combinatorics | 82 |

Bethes States for Generalized XXXand XXZ models | 151 |

Transition on Grothendieck Polynomials | 164 |

Tableau Representation for Macdonalds Ninth Variation of | 180 |

analogue of the characters of Finite Dimensional | 196 |

Generalized Holders Theorem for Multiple Gamma Function | 220 |

Complete Integrability | 233 |

Green Functions Associated to Complex Reflection Groups | 281 |

Ge ri | 298 |

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action affine symmetric group Berenstein bijection birational Calogero Calogero-Moser models character cocharge coefficients combinatorial compute conjecture conserved quantities Corollary Coxeter group decomposition definition Dehn twist denote diagonal diagram eigenfunctions equation explicit finite dimensional formula gamma function Gelfand-Tsetlin given Green functions Grothendieck polynomials GT-patterns Hamiltonian highest weight Hilbert scheme homomorphism integer inverse involution isomorphism Kirillov Lascoux Laurent Laurent's polynomial Lax pair Lemma Lett Lie algebra Lusztig Macdonald polynomials Math matrix minor determinant monomial multiple gamma function obtain operators permutation Phys piece-wise linear potential preprint proof Proposition rational recursion relation Remark restricted partition function result Robinson-Schensted-Knuth correspondence root systems satisfy Schiitzenberger Schubert Schubert polynomials Schur functions Section semistandard Young tableaux set of semistandard sheaf space of triangles statistics super symmetric functions symmetric group Theorem theory transformations triangles Xn Tropical image tropical version unipotent variables vector Weyl group Young tableaux

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Page 22 - KAZUHIRO HIKAMI Department of Physics, Graduate School of Science University of Tokyo Hongo 7-3-1, Bunkyo, Tokyo 113-0033, Japan E-mail: