Computational Invariant TheorySpringer Science & Business Media, 12 juin 2002 - 268 pages 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. |
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 |
| 247 | |
| 261 | |
| 263 | |
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Expressions et termes fréquents
action acts affine algebra algebraic group algorithm apply assume bases calculated called char(K characteristic choose closed coefficients Cohen-Macaulay compute consider construction contains Corollary defined Definition dimension elements equal Equation Example exists fact field finite groups follows formula function give given graded Gröbner basis group G Hence Hilbert series homogeneous homogeneous system homomorphism ideal integral invariant ring invariant theory irreducible isomorphic Lemma lies linear algebraic linearly reductive Math matrix maximal methods minimal modular module monomial monomial ordering multiple normal obtain orbit permutation points polynomial ring positive primary invariants Proof properties Proposition prove quotient rational reductive group relations Remark representation respect result Reynolds operator secondary invariants separating shows step subgroup Suppose symmetric system of parameters Theorem variety vector space weight write yields zero
Fréquemment cités
Page 247 - Type II codes, even unimodular lattices, and invariant rings IEEE Trans Inf Theory 45, No 4.