Derksen / Kemper Computational Invariant Theory

2. Auflage 2015
ISBN: 978-3-662-48422-7
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 387 Seiten, eBook

Reihe: Encyclopaedia of Mathematical Sciences

This book is about the computational aspects of invariant theory. Of central interest is the question how the  invariant ring of a given group action can be calculated. Algorithms for this purpose form the main pillars around which the book is built. There are two introductory chapters, one on Gröbner basis methods and one on the basic concepts of invariant theory, which prepare the ground for the algorithms. Then algorithms for computing invariants of finite and reductive groups are discussed. Particular emphasis lies on interrelations between structural properties of invariant rings and computational methods. Finally, the book contains a chapter on applications of invariant theory, covering fields as disparate as graph theory, coding theory, dynamical systems, and computer vision.

The book is intended for postgraduate students as well as researchers in geometry, computer algebra, and, of course, invariant theory. The text is enriched with numerous explicit examples which illustrate the theory and should be of more than passing interest.

More than ten years after the first publication of the book, the second edition now provides a major update and covers many recent developments in the field. Among the roughly 100 added pages there are two appendices, authored by Vladimi

r Popov, and an addendum by Norbert A'Campo and Vladimir Popov.

Zielgruppe

Research

Autoren/Hrsg.

Derksen, Harm

Kemper, Gregor

Inhaltsverzeichnis

1;Preface to the Second Edition;8
2;Preface to the First Edition;10
3;Contents;12
4;Introduction;18
4.1;References;22
5;1 Constructive Ideal Theory;24
5.1;1.1 Ideals and Gröbner Bases;25
5.1.1;1.1.1 Monomial Orderings;25
5.1.2;1.1.2 Gröbner Bases;27
5.1.3;1.1.3 Normal Forms;28
5.1.4;1.1.4 The Buchberger Algorithm;29
5.2;1.2 Elimination Ideals;31
5.2.1;1.2.1 Image Closure of Morphisms;32
5.2.2;1.2.2 Relations Between Polynomials;32
5.2.3;1.2.3 The Intersection of Ideals;33
5.2.4;1.2.4 The Colon Ideal;33
5.2.5;1.2.5 The Krull Dimension;34
5.3;1.3 Syzygy Modules;36
5.3.1;1.3.1 Computing Syzygies;36
5.3.2;1.3.2 Free Resolutions;39
5.4;1.4 Hilbert Series;40
5.4.1;1.4.1 Computation of Hilbert Series;43
5.5;1.5 The Radical Ideal;45
5.5.1;1.5.1 Reduction to Dimension Zero;45
5.5.2;1.5.2 Positive Characteristic;46
5.6;1.6 Normalization;47
5.7;References;51
6;2 Invariant Theory;54
6.1;2.1 Invariant Rings;54
6.2;2.2 Reductive Groups;60
6.2.1;2.2.1 Linearly Reductive Groups;61
6.2.2;2.2.2 Other Notions of Reductivity;66
6.3;2.3 Categorical Quotients;68
6.4;2.4 Separating Invariants;71
6.5;2.5 Homogeneous Systems of Parameters;77
6.5.1;2.5.1 Hilbert's Nullcone;77
6.5.2;2.5.2 Existence of Homogeneous Systems of Parameters;79
6.6;2.6 The Cohen-Macaulay Property of Invariant Rings;80
6.6.1;2.6.1 The Cohen-Macaulay Property;80
6.6.2;2.6.2 The Hochster-Roberts Theorem;82
6.7;2.7 Hilbert Series of Invariant Rings;88
6.8;References;90
7;3 Invariant Theory of Finite Groups;94
7.1;3.1 Homogeneous Components;95
7.1.1;3.1.1 The Linear Algebra Method;96
7.1.2;3.1.2 The Reynolds Operator;96
7.2;3.2 Noether's Degree Bound;97
7.3;3.3 Degree Bounds in the Modular Case;101
7.3.1;3.3.1 Richman's Lower Degree Bound;102
7.3.2;3.3.2 Symonds' Degree Bound;105
7.4;3.4 Molien's Formula;106
7.4.1;3.4.1 Characters and Molien's Formula;107
7.4.2;3.4.2 Extensions to the Modular Case;109
7.4.3;3.4.3 Extended Hilbert Series;112
7.5;3.5 Primary Invariants;114
7.5.1;3.5.1 Dade's Algorithm;115
7.5.2;3.5.2 An Algorithm for Optimal Homogeneous Systems Parameters;116
7.5.3;3.5.3 Constraints on the Degrees of Primary Invariants;117
7.6;3.6 Cohen-Macaulayness;120
7.7;3.7 Secondary Invariants;123
7.7.1;3.7.1 The Nonmodular Case;124
7.7.2;3.7.2 The Modular Case;127
7.8;3.8 Minimal Algebra Generators and Syzygies;129
7.8.1;3.8.1 Algebra Generators from Primary and Secondary Invariants;129
7.8.2;3.8.2 Direct Computation of Algebra Generators: King's Algorithm;130
7.8.3;3.8.3 Computing Syzygies;132
7.9;3.9 Properties of Invariant Rings;134
7.9.1;3.9.1 The Cohen-Macaulay Property;134
7.9.2;3.9.2 Free Resolutions and Depth;135
7.9.3;3.9.3 The Hilbert Series;138
7.9.4;3.9.4 Polynomial Invariant Rings and Reflection Groups;138
7.9.5;3.9.5 The Gorenstein Property;143
7.10;3.10 Permutation Groups;146
7.10.1;3.10.1 Direct Products of Symmetric Groups;146
7.10.2;3.10.2 Göbel's Algorithm;148
7.10.3;3.10.3 SAGBI Bases;153
7.11;3.11 Ad Hoc Methods;154
7.11.1;3.11.1 Finding Primary Invariants;155
7.11.2;3.11.2 Finding Secondary Invariants;157
7.11.3;3.11.3 The Other Exceptional Reflection Groups;161
7.12;3.12 Separating Invariants;162
7.12.1;3.12.1 Degree Bounds;162
7.12.2;3.12.2 Polynomial Separating Subalgebras and Reflection Groups;163
7.13;3.13 Actions on Finitely Generated Algebras;165
7.14;References;170
8;4 Invariant Theory of Infinite Groups;176
8.1;4.1 Computing Invariants of Linearly Reductive Groups;176
8.1.1;4.1.1 The Heart of the Algorithm;176
8.1.2;4.1.2 The Input: The Group and the Representation;179
8.1.3;4.1.3 The Algorithm;182
8.2;4.2 Improvements and Generalizations;187
8.2.1;4.2.1 Localization of the Invariant Ring;188
8.2.2;4.2.2 Generalization to Arbitrary Graded Rings;192
8.2.3;4.2.3 Covariants;195
8.3;4.3 Invariants of Tori;197
8.4;4.4 Invariants of SLn and GLn;201
8.4.1;4.4.1 Binary Forms;203
8.5;4.5 The Reynolds Operator;205
8.5.1;4.5.1 The Dual Space K[G]*;207
8.5.2;4.5.2 The Reynolds Operator for Semi-simple Groups;209
8.5.3;4.5.3 Cayley's Omega Process;216
8.6;4.6 Computing Hilbert Series;221
8.6.1;4.6.1 A Generalization of Molien's Formula;221
8.6.2;4.6.2 Hilbert Series of Invariant Rings of Tori;225
8.6.3;4.6.3 Hilbert Series of Invariant Rings of Connected Reductive Groups;227
8.6.4;4.6.4 Hilbert Series and the Residue Theorem;229
8.7;4.7 Degree Bounds for Invariants;239
8.7.1;4.7.1 Degree Bounds for Orbits;242
8.7.2;4.7.2 Degree Bounds for Tori;247
8.8;4.8 Properties of Invariant Rings;249
8.9;4.9 Computing Invariants of Reductive Groups;250
8.9.1;4.9.1 Computing Separating Invariants;251
8.9.2;4.9.2 Computing the Purely Inseparable Closure;255
8.9.3;4.9.3 Actions on Varieties;259
8.10;4.10 Invariant Fields and Localizations of Invariant Rings;263
8.10.1;4.10.1 Extendend Derksen Ideals and CAGEs;264
8.10.2;4.10.2 The Italian Problem;268
8.10.3;4.10.3 Geometric Aspects of Extended Derksen Ideals;269
8.10.4;4.10.4 Computational Aspects of Extended Derksen Ideals;271
8.10.5;4.10.5 The Additive Group;277
8.10.6;4.10.6 Invariant Rings and Quasi-affine Varieties;281
8.11;References;284
9;5 Applications of Invariant Theory;288
9.1;5.1 Cohomology of Finite Groups;288
9.2;5.2 Galois Group Computation;289
9.2.1;5.2.1 Approximating Zeros;292
9.2.2;5.2.2 The Symbolic Approach;293
9.3;5.3 Noether's Problem and Generic Polynomials;295
9.4;5.4 Systems of Algebraic Equations with Symmetries;298
9.5;5.5 Graph Theory;299
9.6;5.6 Combinatorics;301
9.7;5.7 Coding Theory;304
9.8;5.8 Equivariant Dynamical Systems;306
9.9;5.9 Material Science;308
9.10;5.10 Computer Vision;311
9.10.1;5.10.1 View Invariants of 3D Objects;311
9.10.2;5.10.2 Invariants of n Points on a Plane;312
9.10.3;5.10.3 Moment Invariants;314
9.11;References;316
10;A Linear Algebraic Groups;320
10.1;A.1 Linear Algebraic Groups;320
10.2;A.2 The Lie Algebra of a Linear Algebraic Group;322
10.3;A.3 Reductive and Semi-simple Groups;326
10.4;A.4 Roots;327
10.5;A.5 Representation Theory;329
10.6;References;330
11;B Is One of the Two Orbits in the Closure of the Other?;331
11.1;B.1 Introduction;331
11.2;B.2 Examples;332
11.3;B.3 Algorithm;334
11.4;B.4 Defining the Set G·L by Equations;339
11.5;References;343
12;C Stratification of the Nullcone;345
12.1;C.1 Introduction;345
12.2;C.2 The Stratification;347
12.3;C.3 The Algorithm;352
12.4;C.4 Examples;358
12.5;References;365
13;Addendum to Appendix C: The Source Code of HNC;366
13.1;References;379
14;Notation;380
15;Index;382