گروه‌ها، تولیدکننده‌ها، سیزیگی‌ها و مدارها در نظریهٔ invariant

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. 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 Invariant theory has a long history, almost one-and-a-half centuries, and arose from certain problems from number theory, algebra, and geometry in the works of Gauss, Jacobi, Eisenstein, and Hermite. Invariants came into existence as a tool to distinguish and classify nonequivalent objects in the algebraic problems, where the equivalence relation in given by the orbits of a group [italic capital]G acting on a set [italic capital]X. The results presented in this book were included in lectures delivered at various international conferences and seminars from 1976-1983 V.l. Popov ; [translated From The Russian By A. Martsinkovsky]. Translation Of: Gruppy, Obrazui︠u︡shchie, Sizigii I Orbity V Teorii Invariantov. Includes Bibliographical References (p. 231-242) And Index.