The history of invariant theory spans nearly a century and a half, with roots in certain problems from number theory, algebra, and geometry appearing in the work of Gauss, Jacobi, Eisenstein, and Hermite. Although the connection between invariants and orbits was essentially discovered in the work of Aronhold and Boole, a clear understanding of this connection had not been achieved until recently, when invariant theory was in fact subsumed by a general theory of algebraic groups.
Written by one of the major leaders in the field, this book provides an excellent, comprehensive exposition of invariant theory. Its point of view is unique in that it combines both modern and classical approaches to the subject. The introductory chapter sets the historical stage for the subject, helping to make the book accessible to nonspecialists.
Readership: Graduate students and research mathematicians interested in invariant theory.
Author(s): Vladimir Leonidovich Popov
Series: Translations of Mathematical Monographs, Vol. 100
Publisher: American Mathematical Society
Year: 1992
Language: English
Pages: C+vi+245+B
Cover
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
Back Cover
