Lorenz Multiplicative Invariant Theory

1. Auflage 2005
ISBN: 978-3-540-27358-5
Verlag: Springer-Verlag
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 180 Seiten

Multiplicative invariant theory, as a research area in its own right within the wider spectrum of invariant theory, is of relatively recent vintage. The present text offers a coherent account of the basic results achieved thus far..

Multiplicative invariant theory is intimately tied to integral representations of finite groups. Therefore, the field has a predominantly discrete, algebraic flavor. Geometry, specifically the theory of algebraic groups, enters through Weyl groups and their root lattices as well as via character lattices of algebraic tori.

Throughout the text, numerous explicit examples of multiplicative invariant algebras and fields are presented, including the complete list of all multiplicative invariant algebras for lattices of rank 2.

The book is intended for graduate and postgraduate students as well as researchers in integral representation theory, commutative algebra and, mostly, invariant theory.

Autoren/Hrsg.

Lorenz, Martin

Inhaltsverzeichnis

1;Preface;6
2;Contents;8
3;Introduction;11
3.1;The Setting;11
3.2;Multiplicative and Polynomial Invariants;12
3.3;Special Features of Multiplicative Actions;13
3.4;Origins and Uses of Multiplicative Invariants;14
3.5;Overview of the Contents;15
4;Notations and Conventions;18
5;1 Groups Acting on Lattices;21
5.1;1.1 Introduction;21
5.2;1.2 G-Lattices;21
5.3;1.3 Examples;22
5.4;1.4 Standard Lattice Constructions;24
5.5;1.5 Indecomposable Lattices;26
5.6;1.6 Conditioning the Lattice;28
5.7;1.7 Reflections and Generalized Reflections;29
5.8;1.8 Lattices Associated with Root Systems;32
5.9;1.9 The Root System Associated with a Faithful G-Lattice;35
5.10;1.10 Finite Subgroups of GLn(Z);36
6;2 Permutation Lattices and Flasque Equivalence;40
6.1;2.1 Introduction;40
6.2;2.2 Permutation Lattices;40
6.3;2.3 Stable Permutation Equivalence;41
6.4;2.4 Permutation Projective Lattices;42
6.5;2.5 Hi-trivial, Flasque and Coflasque Lattices;43
6.6;2.6 Flasque and Coflasque Resolutions;44
6.7;2.7 Flasque Equivalence;45
6.8;2.8 Quasi-permutation Lattices and Monomial Lattices;46
6.9;2.9 An Invariant for Flasque Equivalence;47
6.10;2.10 Overview of Lattice Types;49
6.11;2.11 Restriction to the Sylow Normalizer;50
6.12;2.12 Some Sn - Lattices;51
7;3 Multiplicative Actions;58
7.1;3.1 Introduction;58
7.2;3.2 The Group Algebra of a G-Lattice;58
7.3;3.3 Reduction to Finite Groups, Z -structure, and Finite Generation;59
7.4;3.4 Units and Semigroup Algebras;60
7.5;3.5 Examples;62
7.6;3.6 Multiplicative Invariants ofWeight Lattices;68
7.7;3.7 Passage to an Effective Lattice;70
7.8;3.8 Twisted Multiplicative Actions;71
7.9;3.9 Hopf Structure;73
7.10;3.10 Torus Invariants;74
8;4 Class Group;75
8.1;4.1 Introduction;75
8.2;4.2 Some Examples;76
8.3;4.3 Krull Domains and Class Groups;76
8.4;4.4 Samuel’s Exact Sequence;78
8.5;4.5 Generalized Reflections on Rings;79
8.6;4.6 Proof of Theorem 4.1.1;81
9;5 Picard Group;82
9.1;5.1 Introduction;82
9.2;5.2 Invertible Modules;83
9.3;5.3 The Skew Group Ring;84
9.4;5.4 The Trace Map;85
9.5;5.5 The Kernel of the Map Pic(RG) . Pic(R);86
9.6;5.6 The Case of Multiplicative Actions;88
10;6 Multiplicative Invariants of Reflection Groups;90
10.1;6.1 Introduction;90
10.2;6.2 Proof of Theorem 6.1.1;91
10.3;6.3 Computing the Ring of Invariants;92
10.4;6.4 SAGBI Bases;96
11;7 Regularity;100
11.1;7.1 Introduction;100
11.2;7.2 Projectivity over Invariants;101
11.3;7.3 Linearization by the Slice Method;102
11.4;7.4 Proof of Theorem 7.1.1;104
11.5;7.5 Regularity at the Identity;105
12;8 The Cohen-Macaulay Property;107
12.1;8.1 Introduction;107
12.2;8.2 Height and Grade;108
12.3;8.3 Local Cohomology;109
12.4;8.4 Cohen-Macaulay Modules and Rings;110
12.5;8.5 The Cohen-Macaulay Property for Invariant Rings;111
12.6;8.6 The Ellingsrud-Skjelbred Spectral Sequences;114
12.7;8.7 Annihilators of Cohomology Classes;116
12.8;8.8 The Restriction Map for Cohen-Macaulay Invariants;118
12.9;8.9 The Case of Multiplicative Invariants;120
12.10;8.10 Proof of Theorem 8.1.1;122
12.11;8.11 Examples;125
13;9 Multiplicative Invariant Fields;128
13.1;9.1 Introduction;128
13.2;9.2 Stable Isomorphism;131
13.3;9.3 Retract Rationality;131
13.4;9.4 The “No-name Lemma";134
13.5;9.5 Function Fields of Algebraic Tori;136
13.6;9.6 Some Rationality Results for Multiplicative Invariant Fields;141
13.7;9.7 Some Concepts from Algebraic Geometry;144
13.8;9.8 The Field of Matrix Invariants as a Multiplicative Invariant Field;145
14;10 Problems;152
14.1;10.1 The Cohen-Macaulay Problem;152
14.2;10.2 Semigroup Algebras;154
14.3;10.3 Computational Issues;157
14.4;10.4 Essential Dimension Estimates;158
14.5;10.5 Rationality Problems;162
15;References;164
16;Index;175