Classical Invariant Theory

by Peter J. Olver

• ISBN13: 9780521558211
• ISBN10: 0521558212
• Format: Paperback
• Copyright: 1999-01-13
• Publisher: Cambridge University Press

Summary

There has been a resurgence of interest in classical invariant theory driven by several factors: new theoretical developments; a revival of computational methods coupled with powerful new computer algebra packages; and a wealth of new applications, ranging from number theory to geometry, physics to computer vision. This book provides readers with a self-contained introduction to the classical theory as well as modern developments and applications. The text concentrates on the study of binary forms (polynomials) in characteristic zero, and uses analytical as well as algebraic tools to study and classify invariants, symmetry, equivalence and canonical forms. A variety of innovations make this text of interest even to veterans of the subject; these include the use of differential operators and the transform approach to the symbolic method, extension of results to arbitrary functions, graphical methods for computing identities and Hilbert bases, complete systems of rationally and functionally independent covariants, introduction of Lie group and Lie algebra methods, as well as a new geometrical theory of moving frames and applications. Aimed at advanced undergraduate and graduate students the book includes many exercises and historical details, complete proofs of the fundamental theorems, and a lively and provocative exposition.

Table of Contents

Introduction	x
Notes to the Reader	xi
A Brief History	xviii
Acknowledgments	xxi
Prelude --- Quadratic Polynomials and Quadratic Forms	1	(10)
Quadratic Polynomials	2	(3)
Quadratic Forms and Projective Transformations	5	(6)
Basic Invariant Theory for Binary Forms	11	(33)
Binary Forms	11	(2)
Transformation Rules	13	(2)
The Geometry of Projective Space	15	(4)
Homogeneous Functions and Forms	19	(2)
Roots	21	(3)
Invariants and Covariants	24	(1)
The Simplest Examples	25	(5)
Degree, Order, and Weight	30	(2)
Construction of Covariants	32	(1)
Joint Covariants and Polarization	33	(2)
Resultants and Discriminants	35	(4)
The Hilbert Basis Theorem	39	(2)
Syzygies	41	(3)
Groups and Transformations	44	(18)
Basic Group Theory	44	(3)
Group Homomorphisms	47	(3)
Transfomation Groups	50	(4)
Symmetry Groups, Invariant Sets, and Orbits	54	(4)
Equivalence and Canonical Forms	58	(4)
Representations and Invariants	62	(24)
Representations	62	(4)
Irreducibility	66	(3)
Function Spaces	69	(4)
Invariant Functions	73	(3)
Joint Invariants	76	(3)
Multiplier Representations	79	(4)
Relative Invariants	83	(3)
Transvectants	86	(13)
The Omega Process	87	(3)
Projective Coordinates	90	(2)
Partial Transvectants	92	(4)
The Scaling and Polarization Processes	96	(1)
The Poisson and Moyal Brackets	97	(2)
Symbolic Methods	99	(29)
The Fourier Transform	100	(2)
The General Transform	102	(5)
Brackets	107	(3)
Syzygies	110	(2)
The Classical Symbolic Method	112	(5)
Proofs of the Fundamental Theorems	117	(5)
Reciprocity	122	(2)
Fundamental Systems of Covariants	124	(4)
Graphical Methods	128	(22)
Digraphs, Molecules, and Covariants	129	(4)
Syzygies and the Algebra of Digraphs	133	(4)
Graphical Representation of Transvectants	137	(3)
Transvectants of Homogeneous Functions	140	(3)
Gordan's Method	143	(7)
Lie Groups and Moving Frames	150	(48)
Lie Groups	151	(4)
Lie Transformation Groups	155	(2)
Orbits and Invariance	157	(4)
Normalization	161	(5)
Joint Invariants	166	(3)
Prolongation of Group Actions	169	(2)
Differential Invariants	171	(6)
Differential Invariants for Binary Forms	177	(4)
Equivalence and Signature Curves	181	(4)
Symmetries of Curves	185	(3)
Equivalence and Symmetry of Binary Forms	188	(10)
Infinitesimal Methods	198	(30)
One-Parameter Subgroups	199	(2)
Matrix Lie Algebras	201	(4)
Vector Fields and Orbits	205	(5)
Infinitesimal Invariance	210	(5)
Infinitesimal Multipliers and Relative Invariants	215	(1)
Isobaric and Semi-invariants	216	(4)
The Hilbert Operator	220	(3)
Proof of the Hilbert Basis Theorem	223	(3)
Nullforms	226	(2)
Multivariate Polynomials	228	(19)
Polynomials and Algebraic Curves	229	(2)
Transformations and Covariants	231	(3)
Transvectants	234	(2)
Tensorial Invariants	236	(2)
Symbolic Methods	238	(4)
Biforms	242	(5)
References	247	(13)
Author Index	260	(4)
Subject Index	264