Invariant Theory of Finite Groups
About this Title
Mara D. Neusel, University of Notre Dame, Notre Dame, IN and Larry Smith, University of Göttingen, Göttingen, Germany
Publication: Mathematical Surveys and Monographs
Publication Year 2002: Volume 94
ISBNs: 978-0-8218-4981-1 (print); 978-1-4704-1321-7 (online)
MathSciNet review: MR1869812
MSC: Primary 13A50; Secondary 55S10
The questions that have been at the center of invariant theory since the 19th century have revolved around the following themes: finiteness, computation, and special classes of invariants. This book begins with a survey of many concrete examples chosen from these themes in the algebraic, homological, and combinatorial context. In further chapters, the authors pick one or the other of these questions as a departure point and present the known answers, open problems, and methods and tools needed to obtain these answers. Chapter 2 deals with algebraic finiteness. Chapter 3 deals with combinatorial finiteness. Chapter 4 presents Noetherian finiteness. Chapter 5 addresses homological finiteness. Chapter 6 presents special classes of invariants, which deal with modular invariant theory and its particular problems and features. Chapter 7 collects results for special classes of invariants and coinvariants such as (pseudo) reflection groups and representations of low degree. If the ground field is finite, additional problems appear and are compensated for in part by the emergence of new tools. One of these is the Steenrod algebra, which the authors introduce in Chapter 8 to solve the inverse invariant theory problem, around which the authors have organized the last three chapters.
The book contains numerous examples to illustrate the theory, often of more than passing interest, and an appendix on commutative graded algebra, which provides some of the required basic background. There is an extensive reference list to provide the reader with orientation to the vast literature.
Graduate students and research mathematicians interested in commutative rings, algebras, and algebraic topology.
Table of Contents
- 1. Invariants, their relatives, and problems
- 2. Algebraic finiteness
- 3. Combinatorial finiteness
- 4. Noetherian finiteness
- 5. Homological finiteness
- 6. Modular invariant theory
- 7. Special classes of invariants
- 8. The Steenrod algebra and invariant theory
- 9. Invariant ideals
- 10. Lannes’s T-functor and applications