Computational Invariant TheoryInvariant theory is a subject with a long tradition and an astounding abil ity to rejuvenate itself whenever it reappears on the mathematical stage. Throughout the history of invariant theory, two features of it have always been at the center of attention: computation and applications. This book is about the computational aspects of invariant theory. We present algorithms for calculating the invariant ring of a group that is linearly reductive or fi nite, including the modular case. These algorithms form the central pillars around which the book is built. To prepare the ground for the algorithms, we present Grabner basis methods and some general theory of invariants. Moreover, the algorithms and their behavior depend heavily on structural properties of the invariant ring to be computed. Large parts of the book are devoted to studying such properties. Finally, most of the applications of in variant theory depend on the ability to calculate invariant rings. The last chapter of this book provides a sample of applications inside and outside of mathematics. |
Contents
Introduction | 1 |
Constructive Ideal Theory | 7 |
11 Ideals and Grobner Bases | 8 |
12 Elimination Ideals | 13 |
13 Syzygy Modules | 18 |
14 Hilbert Series | 22 |
15 The Radical Ideal | 27 |
16 Normalization | 32 |
42 Improvements and Generalizations | 150 |
43 Invariants of Tori | 159 |
44 Invariants of SLn and GLn | 162 |
45 The Reynolds Operator | 166 |
46 Computing Hilbert Series | 180 |
47 Degree Bounds for Invariants | 196 |
48 Properties of Invariant Rings | 205 |
5 Applications of Invariant Theory | 209 |
Invariant Theory | 39 |
22 Reductive Groups | 44 |
23 Categorical Quotients | 51 |
24 Homogeneous Systems of Parameters | 59 |
25 The CohenMacaulay Property of Invariant Rings | 62 |
26 Hilbert Series of Invariant Rings | 69 |
Invariant Theory of Finite Groups | 73 |
31 Homogeneous Components | 75 |
32 Moliens Formula | 76 |
33 Primary Invariants | 80 |
34 CohenMacaulayness | 86 |
35 Secondary Invariants | 89 |
36 Minimal Algebra Generators and Syzygies | 95 |
37 Properties of Invariant Rings | 97 |
38 Noethers Degree Bound | 108 |
39 Degree Bounds in the Modular Case | 112 |
310 Permutation Groups | 122 |
311 Ad Hoc Methods | 130 |
Invariant Theory of Reductive Groups | 139 |
52 Galois Group Computation | 210 |
53 Noethers Problem and Generic Polynomials | 215 |
54 Systems of Algebraic Equations with Symmetries | 218 |
55 Graph Theory | 220 |
56 Combinatorics | 222 |
57 Coding Theory | 224 |
58 Equivariant Dynamical Systems | 226 |
59 Material Science | 228 |
510 Computer Vision | 231 |
A Linear Algebraic Groups | 237 |
A2 The Lie Algebra of a Linear Algebraic Group | 239 |
A3 Reductive and Semisimple Groups | 243 |
A4 Roots | 244 |
A5 Representation Theory | 245 |
| 247 | |
| 261 | |
| 263 | |
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Common terms and phrases
A-module affine variety algorithm assume binary forms calculated char(K characteristic coefficients compute coordinate ring Corollary defined deg(fi degree bound dimension elementary symmetric polynomials elements Equation Example exists field finite dimensional finite groups follows G acts G-stable geometric given GL(V graded algebra Gröbner basis group G Hence Hilbert series homogeneous invariants homogeneous system homomorphism ideal integral invariant ring K[V]G invariant theory invariants of degree irreducible isomorphic Kemper kernel KG-module Lemma Let G linear algebraic group linearly reductive group Math matrix maximal minimal modular module Molien's formula monomial ordering morphism Noether's non-zero normal form obtain orbit permutation polynomial ring Popov primary invariants Proof Proposition quotient rational representation Red(t reductive group reflection group representation of G Reynolds operator secondary invariants Section subalgebra subgroup subset symmetric symmetric group system of parameters syzygy Theorem torus vector space Zariski zero
Popular passages
Page 247 - Type II codes, even unimodular lattices, and invariant rings IEEE Trans Inf Theory 45, No 4.
References to this book
Self-Dual Codes and Invariant Theory Gabriele Nebe,Eric M. Rains,Neil J. A. Sloane Limited preview - 2006 |
Computing in Algebraic Geometry: A Quick Start using SINGULAR Wolfram Decker,Christoph Lossen No preview available - 2006 |