Computational Invariant Theory (eBook)

Preface to the Second Edition 8
Preface to the First Edition 10
Contents 12
Introduction 18
References 22
1 Constructive Ideal Theory 24
1.1 Ideals and Gröbner Bases 25
1.1.1 Monomial Orderings 25
1.1.2 Gröbner Bases 27
1.1.3 Normal Forms 28
1.1.4 The Buchberger Algorithm 29
1.2 Elimination Ideals 31
1.2.1 Image Closure of Morphisms 32
1.2.2 Relations Between Polynomials 32
1.2.3 The Intersection of Ideals 33
1.2.4 The Colon Ideal 33
1.2.5 The Krull Dimension 34
1.3 Syzygy Modules 36
1.3.1 Computing Syzygies 36
1.3.2 Free Resolutions 39
1.4 Hilbert Series 40
1.4.1 Computation of Hilbert Series 43
1.5 The Radical Ideal 45
1.5.1 Reduction to Dimension Zero 45
1.5.2 Positive Characteristic 46
1.6 Normalization 47
References 51
2 Invariant Theory 54
2.1 Invariant Rings 54
2.2 Reductive Groups 60
2.2.1 Linearly Reductive Groups 61
2.2.2 Other Notions of Reductivity 66
2.3 Categorical Quotients 68
2.4 Separating Invariants 71
2.5 Homogeneous Systems of Parameters 77
2.5.1 Hilbert's Nullcone 77
2.5.2 Existence of Homogeneous Systems of Parameters 79
2.6 The Cohen-Macaulay Property of Invariant Rings 80
2.6.1 The Cohen-Macaulay Property 80
2.6.2 The Hochster-Roberts Theorem 82
2.7 Hilbert Series of Invariant Rings 88
References 90
3 Invariant Theory of Finite Groups 94
3.1 Homogeneous Components 95
3.1.1 The Linear Algebra Method 96
3.1.2 The Reynolds Operator 96
3.2 Noether's Degree Bound 97
3.3 Degree Bounds in the Modular Case 101
3.3.1 Richman's Lower Degree Bound 102
3.3.2 Symonds' Degree Bound 105
3.4 Molien's Formula 106
3.4.1 Characters and Molien's Formula 107
3.4.2 Extensions to the Modular Case 109
3.4.3 Extended Hilbert Series 112
3.5 Primary Invariants 114
3.5.1 Dade's Algorithm 115
3.5.2 An Algorithm for Optimal Homogeneous Systems Parameters 116
3.5.3 Constraints on the Degrees of Primary Invariants 117
3.6 Cohen-Macaulayness 120
3.7 Secondary Invariants 123
3.7.1 The Nonmodular Case 124
3.7.2 The Modular Case 127
3.8 Minimal Algebra Generators and Syzygies 129
3.8.1 Algebra Generators from Primary and Secondary Invariants 129
3.8.2 Direct Computation of Algebra Generators: King's Algorithm 130
3.8.3 Computing Syzygies 132
3.9 Properties of Invariant Rings 134
3.9.1 The Cohen-Macaulay Property 134
3.9.2 Free Resolutions and Depth 135
3.9.3 The Hilbert Series 138
3.9.4 Polynomial Invariant Rings and Reflection Groups 138
3.9.5 The Gorenstein Property 143
3.10 Permutation Groups 146
3.10.1 Direct Products of Symmetric Groups 146
3.10.2 Göbel's Algorithm 148
3.10.3 SAGBI Bases 153
3.11 Ad Hoc Methods 154
3.11.1 Finding Primary Invariants 155
3.11.2 Finding Secondary Invariants 157
3.11.3 The Other Exceptional Reflection Groups 161
3.12 Separating Invariants 162
3.12.1 Degree Bounds 162
3.12.2 Polynomial Separating Subalgebras and Reflection Groups 163
3.13 Actions on Finitely Generated Algebras 165
References 170
4 Invariant Theory of Infinite Groups 176
4.1 Computing Invariants of Linearly Reductive Groups 176
4.1.1 The Heart of the Algorithm 176
4.1.2 The Input: The Group and the Representation 179
4.1.3 The Algorithm 182
4.2 Improvements and Generalizations 187
4.2.1 Localization of the Invariant Ring 188
4.2.2 Generalization to Arbitrary Graded Rings 192
4.2.3 Covariants 195
4.3 Invariants of Tori 197
4.4 Invariants of SLn and GLn 201
4.4.1 Binary Forms 203
4.5 The Reynolds Operator 205
4.5.1 The Dual Space K[G]* 207
4.5.2 The Reynolds Operator for Semi-simple Groups 209
4.5.3 Cayley's Omega Process 216
4.6 Computing Hilbert Series 221
4.6.1 A Generalization of Molien's Formula 221
4.6.2 Hilbert Series of Invariant Rings of Tori 225
4.6.3 Hilbert Series of Invariant Rings of Connected Reductive Groups 227
4.6.4 Hilbert Series and the Residue Theorem 229
4.7 Degree Bounds for Invariants 239
4.7.1 Degree Bounds for Orbits 242
4.7.2 Degree Bounds for Tori 247
4.8 Properties of Invariant Rings 249
4.9 Computing Invariants of Reductive Groups 250
4.9.1 Computing Separating Invariants 251
4.9.2 Computing the Purely Inseparable Closure 255
4.9.3 Actions on Varieties 259
4.10 Invariant Fields and Localizations of Invariant Rings 263
4.10.1 Extendend Derksen Ideals and CAGEs 264
4.10.2 The Italian Problem 268
4.10.3 Geometric Aspects of Extended Derksen Ideals 269
4.10.4 Computational Aspects of Extended Derksen Ideals 271
4.10.5 The Additive Group 277
4.10.6 Invariant Rings and Quasi-affine Varieties 281
References 284
5 Applications of Invariant Theory 288
5.1 Cohomology of Finite Groups 288
5.2 Galois Group Computation 289
5.2.1 Approximating Zeros 292
5.2.2 The Symbolic Approach 293
5.3 Noether's Problem and Generic Polynomials 295
5.4 Systems of Algebraic Equations with Symmetries 298
5.5 Graph Theory 299
5.6 Combinatorics 301
5.7 Coding Theory 304
5.8 Equivariant Dynamical Systems 306
5.9 Material Science 308
5.10 Computer Vision 311
5.10.1 View Invariants of 3D Objects 311
5.10.2 Invariants of n Points on a Plane 312
5.10.3 Moment Invariants 314
References 316
A Linear Algebraic Groups 320
A.1 Linear Algebraic Groups 320
A.2 The Lie Algebra of a Linear Algebraic Group 322
A.3 Reductive and Semi-simple Groups 326
A.4 Roots 327
A.5 Representation Theory 329
References 330
B Is One of the Two Orbits in the Closure of the Other? 331
B.1 Introduction 331
B.2 Examples 332
B.3 Algorithm 334
B.4 Defining the Set G·L by Equations 339
References 343
C Stratification of the Nullcone 345
C.1 Introduction 345
C.2 The Stratification 347
C.3 The Algorithm 352
C.4 Examples 358
References 365
Addendum to Appendix C: The Source Code of HNC 366
References 379
Notation 380
Index 382