This book presents the characteristic zero invariant theory of finite groups acting linearly on polynomial algebras. The author assumes basic knowledge of groups and rings, and introduces more advanced methods from commutative algebra along the way. The theory is illustrated by numerous examples and applications to physics, engineering, numerical analysis, combinatorics, coding theory, and graph theory. A wide selection of exercises and suggestions for further reading makes the book appropriate for an advanced undergraduate or first-year graduate level course.
Mara D. Neusel : Texas Tech University, Lubbock, TX
1. Introduction
Part 1. Recollections
Chapter 1. Linear representations of finite groups Chapter 2. Rings and algebras
Part 2. Introduction and Göbel’s bound Chapter 3. Rings of polynomial invariants Chapter 4. Permutation representations Application: Decay of a spinless particle Application: Counting weighted graphs
Part 3. The first fundamental theorem of invariant theory and Noether’s bound Chapter 5. Construction of invariants Chapter 6. Noether’s bound Chapter 7. Some families of invariants Application: Production of fibre composites Application: Gaussian quadrature
Part 4. Noether’s theorems Chapter 8. Modules Chapter 9. Integral dependence and the Krull relations Chapter 10. Noether’s theorems Application: Self-dual codes
Part 5. Advanced counting methods and the Shephard-Todd-Chevalley theorem Chapter 11. Poincaré series Chapter 12. Systems of parameters Chapter 13. Pseudoreflection representations Application: Counting partitions Appendix A. Rational invariants