Physics and combinatorics 2000 : proceedings of the Nagoya 2000 International Workshop : Graduate School of Mathematics, Nagoya University, 21-26 August, 2000

The 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 these fields and to discuss existing and possible interactions between the mentioned subjects. This volume constitutes the proceedings of the workshop. Preface......Page 6 CONTENTS......Page 8 1 Introduction......Page 10 2 The Bridgeland-King-Reid theorem......Page 14 3 Prior results on Hilbert schemes and polygraphs......Page 17 4 Main results......Page 19 5 Application to character formulas......Page 24 References......Page 29 1 Introduction......Page 31 2 Ideal g-on Gas and Recursion Relation......Page 32 3 Generalization: Yangian Symmetry......Page 41 4 Concluding Remarks......Page 49 References......Page 55 1 Introduction......Page 58 2 Forrester's conjecture......Page 59 3 q-analogue......Page 64 References......Page 71 1 Introduction......Page 72 2 Quantum dilogarithm......Page 73 3 The basic algebraic system......Page 75 4 The mapping class group representation......Page 77 5 Diagonalizing the Dehn twist......Page 81 6 Braiding and R-matrix......Page 83 7 Conclusion......Page 87 References......Page 89 Introduction to Tropical Combinatorics......Page 91 1 Introduction......Page 92 2 Basic Definitions......Page 99 3 Explicit formulae: piece-wise linear version......Page 121 4 Explicit formulae: tropification......Page 124 References......Page 157 1 Introduction......Page 160 2 Rogers-Ramanujan's type identity......Page 161 3 XXZ -> XXX bijection......Page 167 References......Page 172 1 Introduction......Page 173 2 0-Hecke algebra......Page 175 4 Stable part by restriction or symmetrization.......Page 178 5 Transition on Grothendieck polynomials......Page 181 6 Variations among determinantal expressions......Page 185 References......Page 187 Introduction......Page 189 1 Weyl formula and Jacobi-Trudi formula......Page 190 2 Tableau representation......Page 195 3 Specifying the tableau variables......Page 200 References......Page 204 1 The quantum loop algebra......Page 205 2 Standard modules......Page 208 3 t-analogues of q-characters......Page 210 4 Step 3......Page 211 5 Step 1......Page 212 6 Step 2......Page 215 7 Restriction to Uc(g)......Page 218 8 Quiver varieties......Page 220 References......Page 227 1 Introduction......Page 229 2 Barnes' Multiple Gamma Function......Page 230 3 Generalized Holder's Theorem......Page 232 4 Differential Relation in a Special Case......Page 238 References......Page 240 1 Introduction......Page 242 2 Quantum Calogero-Moser Models......Page 247 3 Coxeter invariant excited states triangularity and spectrum......Page 252 4 Quantum Lax pair and quantum conserved quantities......Page 262 5 Algebraic construction of excited states I......Page 267 6 l operators......Page 270 7 Algebraic construction of excited states II......Page 273 8 Universal proof of involution of quantum conserved quantities for type I II and III models......Page 278 9 Summary comments and outlook......Page 280 References......Page 285 0 Introduction......Page 290 1 A background from finite reductive groups......Page 292 2 Green functions associated to G(e 1 n)......Page 294 3 A combinatorial setting for G(e 1 n)......Page 296 References......Page 307 1 Introduction......Page 308 2 Derivation......Page 309 References......Page 313 1 Introduction......Page 314 2 A birational map R......Page 315 3 Combinatorial R-matrix......Page 320 4 Lusztig's bijection Ri'i......Page 322 5 Discrete Toda equation......Page 324 References......Page 326 The 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 The Nagoya 2000 International Workshop gathered scientists working in combinatorics, representation theory, special functions, number theory and mathematical physics. These papers present some basic results in these fields and discuss existing and possible interactions between the subjects. In an earlier paper, we showed that the Hilbert scheme of points in the plane Hn = Hilbn (C2) can be identified with the Hilbert scheme of regular orbits C2n // Sn for the action of Sn permuting the factors in the Cartesian product C2n = (C2)n.