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Invariant Theory of Finite Groups
Mara D. Neusel, University of Notre Dame, IN, and Larry Smith, University of Göttingen, Germany

Mathematical Surveys and Monographs
2002; 371 pp; softcover
Volume: 94
ISBN-10: 0-8218-4981-6
ISBN-13: 978-0-8218-4981-1
List Price: US$94
Member Price: US$75.20
Order Code: SURV/94.S
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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.


"[The book] covers a lot of information and various instructive examples."

-- Zentralblatt MATH

"Both the material and the treatment would be ideal for a postgraduate course, or for inclusion in ... [a] postgraduate `crash course' dealing with topics in modern algebra ... In addition to recommending this book to all who want to learn about invariant theory, I also recommend it to those in search of a scholium on typography (to be found on pages 357 and 358), which introduces the reader to such esoterica as Zapfian italics!"

-- Bulletin of the LMS

Table of Contents

  • Invariants, their relatives, and problems
  • Algebraic finiteness
  • Combinatorial finiteness
  • Noetherian finiteness
  • Homological finiteness
  • Modular invariant theory
  • Special classes of invariants
  • The Steenrod algebra and invariant theory
  • Invariant ideals
  • Lannes's T-functor and applications
  • Review of commutative algebra
  • References
  • Typography
  • Notation
  • Index
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