Cover......Page 1
Symmetries, Integrable Systems and Representations......Page 4
Preface......Page 7
Contents......Page 8
1 Introduction......Page 10
2.1 Symmetric Functions and Sekiguchi Operators......Page 11
2.3 Grading and Filtration......Page 12
2.4 The Algebra SHc......Page 13
3.2 Realization as a Shuffle Algebra......Page 14
4.1 Generators and Relations for SH+......Page 15
4.2 Generators and Relations for SHc......Page 16
5.2 Verification in Ranks One and Two......Page 17
5.3 The Order Filtration on SH>......Page 18
5.5 The Degree Zero Component......Page 19
5.6 Completion of the Induction Step......Page 20
References......Page 22
1 Introduction......Page 23
1.1 Combinatorial Identities......Page 25
1.2 Cyclic Quiver Varieties......Page 26
1.3 C*-Action on M(v,w)......Page 27
1.4 Quasihomogeneous Components in the Moduli Space of Sheaves......Page 28
2.1 Grothendieck Ring of Quasiprojective Varieties......Page 29
2.2 Proof of Theorem 4......Page 30
3 Proof of Theorem 1......Page 33
3.2 Cores and Quotients......Page 34
3.4 Proof of Theorem 1......Page 35
4 Proofs of Theorems 2 and 3......Page 38
5 Proof of Theorem 5......Page 39
References......Page 40
1 Introduction......Page 42
2 The Setup over the Complex Numbers: Definitions and Notation......Page 44
3 The Kostant Lattice......Page 46
4 Roots and Relations in Type A and C......Page 50
5 The Spanning Property for SLn+1......Page 51
7 Symplectic Dyck Paths......Page 59
8 The Spanning Property for the Symplectic Lie Algebra......Page 60
9 The Tensor Product Property......Page 69
References......Page 70
2 Affine Kac-Moody Algebras......Page 71
2.1 Affine Roots......Page 72
2.3 The Weyl Group......Page 73
3.1 Projective Objects in O......Page 74
3.3 The Level......Page 76
3.4 The Structure of Equivalence Classes......Page 77
4 Extensions of Neighbouring Verma Modules......Page 78
5 Restricted Critical Level Representations......Page 80
5.2 Restricted Representations......Page 81
5.3 Restricted Projective Objects......Page 82
6 The Structure of Subgeneric Critical Restricted Blocks......Page 83
6.2 Multiplicities in the Subgeneric Case......Page 84
6.3 The Partial Restriction Functor......Page 85
6.4 Homomorphisms Between Projectives......Page 87
References......Page 90
1 Overview......Page 91
1.2 Conformal Invariance and 2-Point Functions......Page 92
1.4 Aims of This Work......Page 93
1.6 Non-extremal Operators......Page 94
1.8 Remark......Page 95
2.1 Integrability in AdS/CFT......Page 96
2.3 SYM4 and Spin Chains. 1-Loop Results......Page 97
3.2 Initial Spin-Up and Spin-Down Reference States......Page 98
3.5 The Hamiltonian H......Page 99
3.7 The L-Matrix......Page 100
3.8 The Monodromy Matrix M......Page 101
3.9 The Transfer Matrix T......Page 102
3.10 Generic States, Eigenstates and Bethe Equations......Page 103
3.11 Scalar Products that Are Determinants......Page 104
3.13 The Slavnov Scalar Product S[L,N1,N1]......Page 105
3.15 The Homogeneous Limit of S[L,N1,N2]......Page 106
4 The Trigonometric Six-Vertex Model......Page 107
4.3 Six Vertices that Conserve `Arrow Flow'......Page 108
4.4 Correspondence with the XXZ R-Matrix......Page 109
4.7 Rows of Segments, Spin Systems, Spin System States and Net Spin......Page 110
4.11 Four Types of Horizontal Lines......Page 111
4.13 Four Types of Configurations......Page 112
4.15 Correspondence with S[L,N1,N1] Scalar Products and S[L,N1,N2] Restricted Scalar Products......Page 113
4.16 [L, N1,N2]-Configurations as Restrictions of BC-Configurations......Page 114
4.18 Izergin's Domain Wall Partition Function......Page 117
5.1 Tree-Level Structure Constants......Page 118
5.4 From Single-Trace Operators to Spin-Chain States......Page 119
5.6 Structure Constants in Terms of Spin-Chains......Page 120
5.6.3 Step 3. Compute Scalar Products......Page 121
5.8 Type-A. Simplifying the Unevaluated Expression......Page 122
5.9.1 Step 1. Re-writing One of the Scalar Products......Page 123
5.9.2 Step 2. The Domain Wall Partition Functions......Page 124
6 Structure Constants in Type-B Theories......Page 125
6.2 Similarities Between Type-A and Type-B Theories......Page 126
6.4 One of the Operators Must Be BPS-Like......Page 127
7.1 Notation Related to Sets of Variables......Page 128
7.5 The Discrete KP Hierarchy......Page 129
7.6 Casoratian Matrices and Determinants......Page 130
7.8 Notation for Determinants with Elements omegaij......Page 131
7.10 Casoratians Are Discrete KP tau-Functions......Page 132
7.11 Change of Variables......Page 133
7.13 The Slavnov Scalar Product is a Discrete KP tau-Function......Page 134
8 Summary and Comments......Page 135
References......Page 136
1 Introduction......Page 139
2.2 Centralizer Algebras and the Isomorphism of Bezrukavnikov-Etingof......Page 141
2.3 Category O and Parabolic Restriction and Induction......Page 142
2.4 Basechange......Page 143
2.5 Holomorphic Version......Page 145
3.1 Fundamental Groups......Page 148
3.3 Monodromy......Page 149
3.4 Decomposition of Induction and Restriction......Page 151
3.5 Transitivity......Page 152
4.2 Induction and Restriction......Page 154
4.3 sle-Categorification......Page 156
4.4 Monodromy and the KZ Functor......Page 157
4.5 The KZ-Component of the Crystal......Page 159
References......Page 160
1 Introduction......Page 161
2 Context......Page 162
3 Summary of [7] and [8] in the osp(2m+1,2n) Case......Page 164
Important Remark......Page 165
3.1 Summary of [7] for osp(2n+1,2n)......Page 166
3.2 Summary of [8] for osp(2m+1,2n)......Page 167
4 Computing Characters for a Simple Maximally Atypical Module over osp(5,4)......Page 169
5 Projective Indecomposable Modules for osp(5,4), Maximally Atypical Case......Page 173
6 Generic Picture for osp(7,6), Exceptional Moves for osp(7,6) (and Remarks on Higher Rank Cases)......Page 177
6.2 Exceptional Moves......Page 178
References......Page 179
1 Introduction......Page 180
2.1 Cluster Algebras......Page 181
2.2 Monoidal Categorifications......Page 182
3.2 Quantum Loop Algebra......Page 183
3.3 The Monoidal Category Cxi......Page 184
3.4 Restriction and Decomposition......Page 185
4.1 A Cluster Algebra of Type A......Page 186
4.2 Cluster Structure on Cxi......Page 187
5.1 A Cluster Algebra of Type D......Page 189
5.2 Cluster Structure on Cxi......Page 192
References......Page 197
1 Introduction......Page 199
2.1 Symmetric Kac-Moody Root Systems......Page 202
2.2 Basic Roots with a Fixed Index......Page 204
2.4 Basic Roots with Index -2......Page 210
3.1.1 Local Structures......Page 219
3.1.2 Spectral Types and the Euler Transform......Page 222
3.2.1 The Lattice of Spectral Types......Page 226
3.2.2 The Lattice of Spectral Types as a Quotient Lattice......Page 228
3.2.3 Phi-Root System......Page 229
3.3 A Classification of Basic Pairs......Page 230
3.3.1 The Finiteness of Basic Pairs......Page 236
3.3.2 The Classification of Basic Pairs with idx0......Page 237
3.3.3 The Classification of Basic Pairs with idx-2......Page 239
References......Page 244
1 Introduction......Page 246
2 Transfer Matrix and Q-Matrices......Page 247
3 Quasi-local Operators......Page 250
4 Introducing Fermions......Page 251
Commutation Relations......Page 253
6 Expectation Values......Page 254
7 Concluding Remarks......Page 257
Appendix A: Formula for omega(zeta,xi)......Page 258
Appendix B: Anti-commutativity of Fermionic Creation Operators......Page 259
References......Page 263
2 Kaufman and Lamb......Page 265
3 Correlations and Form Factors......Page 266
4 Jimbo, Miwa and Painlevé......Page 273
5 The Susceptibility......Page 276
5.1 The Amplitude of the Susceptibility Divergence......Page 278
5.2 Nickel Singularities and the Natural Boundary Conjecture......Page 279
5.3.1 Direct Sum Decompositions......Page 280
5.3.2 Singularities......Page 281
6 Diagonal Susceptibility......Page 282
6.1 Integral Representations......Page 283
6.3 Direct Sum Decomposition......Page 284
6.4 Results for chi(3)d(t)......Page 285
6.5 Results for chi(4)d(t)......Page 286
6.7 Singularities and Cancellations......Page 287
7 Conclusion......Page 288
7.2 Form Factors, Exponential Forms and Amplitudes......Page 289
7.3 Exponentiation......Page 291
7.5 Natural Boundaries and lambda Extensions......Page 292
7.6 Row Correlations......Page 293
References......Page 294
1 Introduction......Page 298
2.1 Kac-Moody Algebras and Kac-Moody Groups......Page 300
2.2 Geometric Crystals......Page 301
2.4 Positive Structure, Ultra-Discretizations and Tropicalizations......Page 302
3 Perfect Crystals of Type An(1)......Page 304
4.1 Fundamental Representation W(pi2) for An(1)......Page 307
4.2 Affine Geometric Crystal V(An(1)) in W(pi2)......Page 308
5 Ultra-Discretization of V(An(1))......Page 313
References......Page 318
1 Introduction......Page 320
2 Proof of Theorem A......Page 321
2.1 A Lattice VOA......Page 322
2.2 In the Free Bosonic Fock Space M2(1)......Page 323
2.3 Modulo C2(M2(1)sigma)......Page 324
2.4 A Subring......Page 325
2.5 Elements a(-1)a(-1)a......Page 329
2.6 The Action of gamma(4)......Page 330
2.7 Nilpotency of alpha Modulo C2(VLsigma)......Page 331
2.8 C2-Cofiniteness of VLsigma......Page 334
3 Z3-Orbifold Construction......Page 338
3.1 The Character of the Moonshine VOA......Page 343
References......Page 344
1 Introduction......Page 346
2 Words......Page 348
3 Algebraic Invariants......Page 349
4 Automata......Page 350
5 Motivation and Review......Page 354
References......Page 360
1 Introduction......Page 361
2.1 Preliminaries on Root Data......Page 363
2.2 BZ Data Associated to a Finite Interval......Page 364
2.3 Crystal Structure on BZ Data Associated to a Finite Interval......Page 365
2.4 Lusztig Data vs. BZ Data......Page 366
2.5 BZ Data Arising from the Lagrangian Construction of B(infty)......Page 369
2.6 BZ Data Associated to Z......Page 372
2.7 Action of Kashiwara Operators......Page 374
2.8 BZ Data of Type Al-1(1)......Page 375
2.9 Crystal Structure on BZZsigma......Page 376
3.1 The Operator #......Page 378
3.2 Ordinary Crystal Structure on BZIe......Page 382
4.1 Definition of Ordinary Kashiwara Operators on BZZe......Page 384
4.2 Proof of Proposition 11......Page 385
4.3 Ordinary Crystal Structure on (BZZe)sigma......Page 391
4.4 Uniqueness of an Element of Weight Zero......Page 396
4.5 Some Other Properties......Page 398
5.1 Strategy......Page 400
5.2 Proof of Theorem 5 and the Connectedness......Page 401
References......Page 402
1 Introduction......Page 403
1.1 Notations......Page 404
2 Quiver Varieties......Page 405
3.1 Fixed Points......Page 407
3.2 Review of [17]......Page 409
3.3 The Fiber Product ZT......Page 410
4 Coproduct......Page 411
4.2 Convolution by ZT......Page 412
4.3 Coproduct by Convolution......Page 415
4.4 Sheaf-Theoretic Analysis......Page 416
4.5 Coassociativity......Page 419
4.6 Equivariant Homology Version......Page 421
5.1 Decomposition of the Direct Image Sheaf......Page 422
5.2 A Description of Htop(ZT)......Page 424
5.3 Tensor Product Multiplicities in Terms of IC Sheaves......Page 425
5.4 Fixed Point Version......Page 426
References......Page 427
1 Introduction......Page 429
2 Schur Function......Page 432
3 tau-Function......Page 447
4 sigma-Function......Page 451
5 Addition Formulae......Page 458
References......Page 461
1 Introduction......Page 463
Padé Problem......Page 464
Remark on the Choice of the Bases phii(x), chii(x)......Page 465
Special Direction T of Deformation......Page 466
4 Elliptic Painlevé Equation......Page 471
5 Lax Formalism......Page 473
6 Determinant Formulae......Page 476
Appendix: Affine Weyl Group Actions......Page 479
References......Page 481
1 Introduction......Page 483
2 Inequalities......Page 485
4 Real Picture Versus Complex Picture......Page 487
References......Page 489
1 Introduction......Page 491
2 The Inversion Formula of Polylogarithms......Page 492
3 The Recursive Riemann-Hilbert Problem of Additive Type......Page 493
References......Page 496
1 Introduction......Page 497
2 Bulk and Edge......Page 498
3 Calogero-Sutherland Hamiltonian......Page 502
4.1 Sphere......Page 506
4.2 Cylinder and Torus......Page 507
5 Some Remarks on Viscosity......Page 509
Appendix: Shape of the Droplet......Page 511
References......Page 512
1 Introduction......Page 514
2 Review on Construction of Rational Elliptic Surfaces......Page 516
3 Ordinary Differential Equations on Rational Elliptic Surfaces......Page 528
4 Canonical Forms of Biquadratic Hamiltonians......Page 532
5 Oguiso-Shioda's Classification......Page 535
6 Bäcklund Transformations......Page 537
References......Page 539
1 Introduction......Page 541
1.1 Proof of Theorem 2......Page 545
1.2 Proof of Theorem 3 and Theorem 4......Page 547
2 General Setting......Page 548
3.2 Energy Exchange Model......Page 552
3.3 Zero-Range Processes......Page 556
References......Page 558
1 Introduction......Page 559
2.1 Summation over Indices......Page 560
2.2 Algebraic Structure of Multiple Harmonic Sums......Page 563
2.3 Algebraic Formulation of the Main Theorem......Page 564
2.4 Proof of the Main Theorem......Page 567
References......Page 571
1 Introduction-A Trinity of Dedekind eta-Function......Page 572
2 Borcherds Phi-Function......Page 574
2.2.1 Automorphic Form as a Multicanonical Form on OmegaLambda+......Page 575
2.2.4 Automorphic Form as a Function on LR+iCL+......Page 576
2.3.2 Borcherds Phi-Function at the Level 2 Cusp......Page 577
3.2 Enriques Surfaces......Page 579
4.1 Analytic Torsion......Page 581
4.2 Borcherds Phi-Function as the Analytic Torsion of Enriques Surface......Page 582
5 Resultants and Borcherds Phi-Function: An Algebraic Counter Part......Page 583
5.1 (2,2,2)-Model of an Enriques Surface......Page 584
5.2 An Algebraic Expression of Borcherds Phi-Function......Page 586
5.3 A 4-Parameter Family of Enriques Surfaces Associated to M3,6(C)......Page 588
6.1 The Matsumoto-Sasaki-Yoshida Model......Page 589
6.2 Theta Function on D......Page 590
6.3 The Case of Jacobian Kummer Surfaces......Page 591
7 Some Problems......Page 592
References......Page 593
1 Introduction......Page 595
1.1 Inhomogeneous Eight-Vertex Transfer Matrix......Page 596
1.2 Combinatorial Line......Page 597
1.3 Homogeneous Limit......Page 598
2 Properties of the Ground State Eigenvector......Page 599
2.2 Exchange Relation......Page 600
2.3 Spin Flip......Page 601
2.4 Wheel Condition and Recurrence Relations......Page 602
3.1 Definition......Page 605
3.3 Symmetry......Page 606
3.4 Recurrence Relation......Page 607
3.5 Half-specialization......Page 608
3.6 Solution as Pfaffians......Page 609
3.7 Further Factorization as Determinants......Page 611
3.8 Alternative Determinant Formula......Page 612
3.9 Uniformization......Page 613
4.1 Summary......Page 616
4.2 Linear Relations......Page 617
4.2.2 A Second Order Differential Equation......Page 618
4.2.3 Homogeneous Limit......Page 619
4.3 Bilinear Recurrence Relations......Page 620
5 Conclusion and Prospects......Page 621
Appendix A: The zeta->0 Trigonometric Limit......Page 622
A.1 Pfaffians......Page 623
A.2 Determinants......Page 624
A.3 More Determinants......Page 625
Appendix B: The zeta->1 Limit......Page 626
Appendix C: Proof of Symmetry of H2m......Page 627
Appendix D: Differential Equations......Page 628
References......Page 632