Modular Invariant Theory

1. Auflage 2011
ISBN: 978-3-642-17404-9
Verlag: Springer
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 234 Seiten

This book covers the modular invariant theory of finite groups, the case when the characteristic of the field divides the order of the group, a theory that is more complicated than the study of the classical non-modular case. Largely self-contained, the book develops the theory from its origins up to modern results. It explores many examples, illustrating the theory and its contrast with the better understood non-modular setting. It details techniques for the computation of invariants for many modular representations of finite groups, especially the case of the cyclic group of prime order. It includes detailed examples of many topics as well as a quick survey of the elements of algebraic geometry and commutative algebra as they apply to invariant theory. The book is aimed at both graduate students and researchers—an introduction to many important topics in modern algebra within a concrete setting for the former, an exploration of a fascinating subfield of algebraic geometry for the latter.

Autoren/Hrsg.

Inhaltsverzeichnis

1;Preface;5
2;Contents;6
3;Index of notations;9
4;First Steps;11
4.1;Groups Acting on Vector Spaces and Coordinate Rings;12
4.1.1;V Versus V*;14
4.2;Constructing Invariants;16
4.3;On Structures and Fundamental Questions;17
4.4;Bounds for Generating Sets;17
4.5;On the Structure of K[V]G: The Non-modular Case;18
4.6;Structure of K[V]G: Modular Case;19
4.7;Invariant Fraction Fields;20
4.8;Vector Invariants;21
4.9;Polarization and Restitution;21
4.10;The Role of the Cyclic Group Cp in Characteristic p;26
4.11;Cp Represented on a 2 Dimensional Vector Space in Characteristic p;27
4.12;A Further Example: Cp Represented on 2V2 in Characteristic p;30
4.13;The Vector Invariants of V2;33
5;Elements of Algebraic Geometry and Commutative Algebra;35
5.1;The Zariski Topology;35
5.2;The Topological Space Spec(S);37
5.3;Noetherian Rings;37
5.4;Localization and Fields of Fractions;39
5.5;Integral Extensions;39
5.6;Homogeneous Systems of Parameters;40
5.7;Regular Sequences;41
5.8;Cohen-Macaulay Rings;42
5.9;The Hilbert Series;44
5.10;Graded Nakayama Lemma;45
5.11;Hilbert Syzygy Theorem;46
6;Applications of Commutative Algebra to Invariant Theory;48
6.1;Homogeneous Systems of Parameters;49
6.2;Symmetric Functions;53
6.3;The Dickson Invariants;54
6.4;Upper Triangular Invariants;55
6.5;Noether's Bound;55
6.6;Representations of Modular Groups and Noether's Bound;57
6.7;Molien's Theorem;59
6.7.1;The Hilbert Series of the Regular Representation of the Klein Group;60
6.7.2;The Hilbert Series of the Regular Representation of C4;62
6.8;Rings of Invariants of p-Groups Are Unique Factorization Domains;63
6.9;When the Fixed Point Subspace Is Large;64
7;Examples;67
7.1;The Cyclic Group of Order 2, the Regular Representation;69
7.2;A Diagonal Representation of C2;70
7.3;Fraction Fields of Invariants of p-Groups;70
7.4;The Alternating Group;72
7.5;Invariants of Permutation Groups;73
7.6;Göbel's Theorem;74
7.7;The Ring of Invariants of the Regular Representation of the Klein Group;77
7.8;The Ring of Invariants of the Regular Representation of C4;80
7.9;A 2 Dimensional Representation of C3, p=2;83
7.10;The Three Dimensional Modular Representationof Cp;83
7.10.1;Prior Knowledge of the Hilbert Series;84
7.10.2;Without the Use of the Hilbert Series;86
8;Monomial Orderings and SAGBI Bases;90
8.1;SAGBI Bases;92
8.1.1;Symmetric Polynomials;96
8.2;Finite SAGBI Bases;98
8.3;SAGBI Bases for Permutation Representations;100
9;Block Bases;105
9.1;A Block Basis for the Symmetric Group;107
9.2;Block Bases for p-Groups;109
10;The Cyclic Group Cp;111
10.1;Representations of Cp in Characteristic p;111
10.2;The Cp-Module Structure of F[Vn];116
10.2.1;Sharps and Flats;116
10.3;The Cp-Module Structure of F[V];119
10.4;The First Fundamental Theorem for V2;121
10.4.1;Dyck Paths and Multi-Linear Invariants;123
10.4.2;Proof of Lemma 7.4.3;128
10.5;Integral Invariants;130
10.6;Invariant Fraction Fields and Localized Invariants;136
10.7;Noether Number for Cp;138
10.8;Hilbert Functions;144
11;Polynomial Invariant Rings;146
11.1;Stong's Example;152
11.2;A Counterexample;153
11.3;Irreducible Modular Reflection Groups;154
11.3.1;Reflection Groups;155
11.3.2;Groups Generated by Homologies of Order Greater than 2;156
11.3.3;Groups Generated by Transvections;156
12;The Transfer;157
12.1;The Transfer for Nakajima Groups;168
12.2;Cohen-Macaulay Invariant Rings of p-Groups;174
12.3;Differents in Modular Invariant Theory;177
12.3.1;Construction of the Various Different Ideals;178
13;Invariant Rings via Localization;182
14;Rings of Invariants which are Hypersurfaces;188
15;Separating Invariants;193
15.1;Relation Between K[V]G and Separating Subalgebras;197
15.2;Polynomial Separating Algebras and Serre's Theorem;200
15.3;Polarization and Separating Invariants;203
16;Using SAGBI Bases to Compute Rings of Invariants;206
17;Ladders;212
17.1;Group Cohomology;214
17.2;Cohomology and Invariant Theory;215
18;References;223
19;Index;230