Computational Invariant Theory

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Springer Science & Business Media, 12 juin 2002 - 268 pages
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Invariant 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.
 

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Table des matières

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
References
247
Notation
261
Index
263
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Fréquemment cités

Page 247 - Type II codes, even unimodular lattices, and invariant rings IEEE Trans Inf Theory 45, No 4.

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