This book is about the computational aspects of invariant theory.Then algorithms for computing invariants of finite and reductive groups are discussed.

The text is enriched with numerous explicit examples which illustrate the theoiw and should be of more than passing interest.

This book is about the computational aspects of invariant theory.Of central interest is the question how the invariant ring of a given group action can be calculated. Algorithms for this purpose form the main pillars around which the book is built. There are two introductory chapters, one on GrObner basis methods and one on the basic concepts of invariant theory, which prepare the ground for the algorithms. Then algorithms for computing invariants of finite and reductive groups are discussed. Particular emphasis lies on interrelations between structural properties of invariant rings and computational methods. Finally, the book contains a chapter on applications of invariant theory, covering fields as disparate as graph theory, coding theory, dynamical systems, and computer vision.

The book is intended for postgraduate students as well as researchers in geometry, computer algebra, and, of course, invariant theory. The text is enriched with numerous explicit examples which illustrate the theoiw and should be of more than passing interest.

Introduction

1 Constructive Ideal Theory

 1.1 Ideals and GrSbner Bases

 1.2 Elimination Ideals 

 1.3 Syzygy Modules

 1.4 Hilbert Series

 1.5 The Radical Ideal

 1.6 Normalization

2 Invariant Theory

 2.1 Invariant Rings

 2.2 Reductive Groups

 2.3 Categorical Quotients

 2.4 Homogeneous Systems of Parameters

 2.5 The Cohen-Macaulay Property of Invariant Rings

 2.6 Hilbert Series of Invariant Rings

3 Invariant Theory of Finite Groups

 3.1 Homogeneous Components

 3.2 Molien's Formula..

 3.3 Primary Invariants

 3.4 Cohen-Macaulayness

 3.5 Secondary Invariants

 3.6 Minimal Algebra Generators and Syzygies

 3.7 Properties of Invariant Rings

 3.8 Noether's Degree Bound

 3.9 Degree Bounds in the Modular Case

 3.10 Permutation Groups

 3.11 Ad Hoc Methods

4 Invariant Theory of Reductive Groups

 4.1 Computing Invariants of Linearly Reductive Groups

 4.2 Improvements and Generalizations

 4.3 Invariants of Tori

 4.4 Invariants of SLn and GLn

 4.5 The Reynolds Operator

 4.6 Computing Hilbert Series

 4.7 Degree Bounds for Invariants

 4.8 Properties of Invariant Rings

5 Applications of Invariant Theory

 5.1 Cohomology of Finite Groups

 5.2 Galois Group Computation

 5.3 Noether's Problem and Generic Polynomials

 5.4 Systems of Algebraic Equations with Symmetries

 5.5 Graph Theory

 5.6 Combinatorics

 5.7 Coding Theory

 5.8 Equivariant Dynamical Systems

 5.9 Material Science

 5.10 Computer Vision

A Linear Algebraic Groups

 A.1 Linear Algebraic Groups

 A.2 The Lie Algebra of a Linear Algebraic Group

 A.3 Reductive and Semi-simple Groups

 A.4 Roots

 A.5 Representation Theory

References

Notation

Index