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Modular Invariant Theory

:	H.E.A. Eddy Campbell, David Wehlau
:	Modular Invariant Theory
:	Springer-Verlag
:	9783642174049
:	1
:	CHF 114.00
:

:	Arithmetik, Algebra
:	English

:	234
:	Wasserzeichen
:	PC/MAC/eReader/Tablet
:	PDF

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.

	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