Classical Invariant Theory

Peter J. Olver

Cambridge University Press, 1999

Table of Contents

• Introduction x
• Notes to the Reader xi
• A Brief History xviii
• Acknowledgments xxi

• Chapter 1 Prelude --- Quadratic Polynomials and Quadratic Forms 1
  • Quadratic Polynomials 2
  • Quadratic Forms and Projective Transformations 5

• Chapter 2 Basic Invariant Theory for Binary Forms 11
  • Binary Forms 11
  • Transformation Rules 13
  • The Geometry of Projective Space 15
  • Homogeneous Functions and Forms 19
  • Roots 21
  • Invariants and Covariants 24
  • The Simplest Examples 25
  • Degree, Order, and Weight 30
  • Construction of Covariants 32
  • Joint Covariants and Polarization 33
  • Resultants and Discriminants 35
  • The Hilbert Basis Theorem 39
  • Syzygies 41

• Chapter 3 Groups and Transformations 44
  • Basic Group Theory 44
  • Group Homomorphisms 47
  • Transformation Groups 50
  • Symmetry Groups, Invariant Sets, and Orbits 54
  • Equivalence and Canonical Forms 58

• Chapter 4 Representations and Invariants 62
  • Representations 62
  • Irreducibility 66
  • Function Spaces 69
  • Invariant Functions 73
  • Joint Invariants 76
  • Multiplier Representations 79
  • Relative Invariants 83

• Chapter 5 Transvectants 86
  • The Omega Process 87
  • Projective Coordinates 90
  • Partial Transvectants 92
  • The Scaling and Polarization Processes 96
  • The Poisson and Moyal Brackets 97

• Chapter 6 Symbolic Methods 99
  • The Fourier Transform 100
  • The General Transform 102
  • Brackets 107
  • Syzygies 110
  • The Classical Symbolic Method 112
  • Proofs of the Fundamental Theorems 117
  • Reciprocity 122
  • Fundamental Systems of Covariants 124

• Chapter 7 Graphical Methods 128
  • Digraphs, Molecules, and Covariants 129
  • Syzygies and the Algebra of Digraphs 133
  • Graphical Representation of Transvectants 137
  • Transvectants of Homogeneous Functions 140
  • Gordan's Method 143

• Chapter 8 Lie Groups and Moving Frames 150
  • Lie Groups 151
  • Lie Transformation Groups 155
  • Orbits and Invariance 157
  • Normalization 161
  • Joint Invariants 166
  • Prolongation of Group Actions 169
  • Differential Invariants 171
  • Differential Invariants for Binary Forms 177
  • Equivalence and Signature Curves 181
  • Symmetries of Curves 185
  • Equivalence and Symmetry of Binary Forms 188

• Chapter 9 Infinitesimal Methods 198
  • One-Parameter Subgroups 199
  • Matrix Lie Algebras 201
  • Vector Fields and Orbits 205
  • Infinitesimal Invariance 210
  • Infinitesimal Multipliers and Relative Invariants 215
  • Isobaric and Semi-invariants 216
  • The Hilbert Operator 220
  • Proof of the Hilbert Basis Theorem 223
  • Nullforms 226

• Chapter 10 Multivariate Polynomials 228
  • Polynomials and Algebraic Curves 229
  • Transformations and Covariants 231
  • Transvectants 234
  • Tensorial Invariants 236
  • Symbolic Methods 238
  • Biforms 242

• References 247
• Author Index 260
• Subject Index 264