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Full Description
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.
Contents
Introduction Recollections: Linear representations of finite groups Rings and algebras Introduction and Gobel's bound: Rings of polynomial invariants Permutation representations Application: Decay of a spinless particle Application: Counting weighted graphs The first fundamental theorem of invariant theory and Noether's bound: Construction of invariants Noether's bound Some families of invariants Application: Production of fibre composites Application: Gaussian quadrature Noether's theorems: Modules Integral dependence and the Krull relations Noether's theorems Application: Self-dual codes Advanced counting methods and the Shephard-Todd-Chevalley theorem: Poincare series Systems of parameters Pseudoreflection representations Application: Counting partitions Appendix A: Rational invariants Suggestions for further reading Index.
