Modular Invariant Theory (eBook)

Preface 5
Contents 6
Index of notations 9
First Steps 11
Groups Acting on Vector Spaces and Coordinate Rings 12
V Versus V* 14
Constructing Invariants 16
On Structures and Fundamental Questions 17
Bounds for Generating Sets 17
On the Structure of K[V]G: The Non-modular Case 18
Structure of K[V]G: Modular Case 19
Invariant Fraction Fields 20
Vector Invariants 21
Polarization and Restitution 21
The Role of the Cyclic Group Cp in Characteristic p 26
Cp Represented on a 2 Dimensional Vector Space in Characteristic p 27
A Further Example: Cp Represented on 2V2 in Characteristic p 30
The Vector Invariants of V2 33
Elements of Algebraic Geometry and Commutative Algebra 35
The Zariski Topology 35
The Topological Space Spec(S) 37
Noetherian Rings 37
Localization and Fields of Fractions 39
Integral Extensions 39
Homogeneous Systems of Parameters 40
Regular Sequences 41
Cohen-Macaulay Rings 42
The Hilbert Series 44
Graded Nakayama Lemma 45
Hilbert Syzygy Theorem 46
Applications of Commutative Algebra to Invariant Theory 48
Homogeneous Systems of Parameters 49
Symmetric Functions 53
The Dickson Invariants 54
Upper Triangular Invariants 55
Noether's Bound 55
Representations of Modular Groups and Noether's Bound 57
Molien's Theorem 59
The Hilbert Series of the Regular Representation of the Klein Group 60
The Hilbert Series of the Regular Representation of C4 62
Rings of Invariants of p-Groups Are Unique Factorization Domains 63
When the Fixed Point Subspace Is Large 64
Examples 67
The Cyclic Group of Order 2, the Regular Representation 69
A Diagonal Representation of C2 70
Fraction Fields of Invariants of p-Groups 70
The Alternating Group 72
Invariants of Permutation Groups 73
Göbel's Theorem 74
The Ring of Invariants of the Regular Representation of the Klein Group 77
The Ring of Invariants of the Regular Representation of C4 80
A 2 Dimensional Representation of C3, p=2 83
The Three Dimensional Modular Representationof Cp 83
Prior Knowledge of the Hilbert Series 84
Without the Use of the Hilbert Series 86
Monomial Orderings and SAGBI Bases 90
SAGBI Bases 92
Symmetric Polynomials 96
Finite SAGBI Bases 98
SAGBI Bases for Permutation Representations 100
Block Bases 105
A Block Basis for the Symmetric Group 107
Block Bases for p-Groups 109
The Cyclic Group Cp 111
Representations of Cp in Characteristic p 111
The Cp-Module Structure of F[Vn] 116
Sharps and Flats 116
The Cp-Module Structure of F[V] 119
The First Fundamental Theorem for V2 121
Dyck Paths and Multi-Linear Invariants 123
Proof of Lemma 7.4.3 128
Integral Invariants 130
Invariant Fraction Fields and Localized Invariants 136
Noether Number for Cp 138
Hilbert Functions 144
Polynomial Invariant Rings 146
Stong's Example 152
A Counterexample 153
Irreducible Modular Reflection Groups 154
Reflection Groups 155
Groups Generated by Homologies of Order Greater than 2 156
Groups Generated by Transvections 156
The Transfer 157
The Transfer for Nakajima Groups 168
Cohen-Macaulay Invariant Rings of p-Groups 174
Differents in Modular Invariant Theory 177
Construction of the Various Different Ideals 178
Invariant Rings via Localization 182
Rings of Invariants which are Hypersurfaces 188
Separating Invariants 193
Relation Between K[V]G and Separating Subalgebras 197
Polynomial Separating Algebras and Serre's Theorem 200
Polarization and Separating Invariants 203
Using SAGBI Bases to Compute Rings of Invariants 206
Ladders 212
Group Cohomology 214
Cohomology and Invariant Theory 215
References 223
Index 230