Groups, Generators, Syzygies, and Orbits in Invariant Theory

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Translations of Mathematical Monographs 100

S Title

Groups, Generators, Syzygies, and Orbits in Invariant Theory

Copyright (c)1992 by the American Mathematical Society
     ISBN 0-8218-4557-8
     QA201.P6613 1992 512\'.944-dc20
     LCCN 92-10604 CIP

Contents

Introduction

Notation and Terminology

CHAPTER 1  The Role of Reductive Groups in Invariant Theory
     §1. Reductive groups and the generalized Hilbert\'s 14th problem
     §2. Quasihomogeneous varieties of reductive groups and the original Hilbert\'s 14th problem

CHAPTER 2  Constructive Invariant Theory
     §1. Formulation and reduction of the problem
     §2. A bound on the degree of a system of parameters and the main theorem
     §3. The radical of the ideal I and the approach suggested by Dieudonne and Carrell

CHAPTER 3  The Degree of the Poincare Series of the Algebra of Invariants and a Finiteness Theorem for Representations with Free Algebra of Invariants
     §1. The degree of the Poincare series and a functional equation
     §2. The zonohedron of weights
     §3. Finiteness theorems

CHAPTER 4  Syzygies in Invariant Theory
     §0. A description of the results and additional notation
     §1. Monotonicity theorems
     §2. Bounds on hd k [ V ]G for certain types of groups
     §3. Estimating hd k [ V ]G with the aid of one-dimensional tori of G
     §4. Majorizing theorems for multiplicities, generic stabilizers, and stability
     §5. Torus T for the classical simple groups of rank > 2
     §6. Torus T for the exceptional simple groups
     §7. Proof of the main theorem: the first case
     §8. Proof of the main theorem: the second case
     §9. Proof of the main theorem: the third case
     §10. Examples

CHAPTER 5  Representations with Free Modules of Covariants
     §1. Connections with equidimensionality: finiteness theorems
     §2. Classification and equivalent characterizations: Igusa\'s condition

CHAPTER 6  A Classification of Normal Affine Quasihomogeneous Varieties of SL2
     §1. Some general results and the beginning of classificatio
     §2. The conclusion of classification
     §3. Application: the structure of orbit closures in finite-dimensional rational SL2-modules

CHAPTER 7  Quasihomogeneous Curves, Surfaces, and Solids
     §1. A classification of irreducible quasihomogeneous curves
     §2. A classification of irreducible afline surfaces with algebraic groups of automorphisms acting transitively on the complement of a finite number of points
     §3. A classification of irreducible affine solids with algebraic groups of automorphisms acting transitively on the complement of a finite number of points

Appendix
     §1. Appendix to Chapter 1
     §2. Appendix to Chapter 2
     §3. Appendix to Chapter 3
     §4. Appendix to Chapter 4
     §5. Appendix to Chapter 5
     §6. Appendix to Chapter 6

Bibliography
     References for the Appendix

Subject Index

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