**Mathematical Surveys and Monographs**

Volume: 94;
2002;
371 pp;
Softcover

MSC: Primary 13; 55;

**Print ISBN: 978-0-8218-4981-1
Product Code: SURV/94.S**

List Price: $95.00

AMS Member Price: $76.00

MAA Member Price: $85.50

**Electronic ISBN: 978-1-4704-1321-7
Product Code: SURV/94.S.E**

List Price: $94.00

AMS Member Price: $75.20

MAA Member Price: $84.60

#### Supplemental Materials

# Invariant Theory of Finite Groups

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*Mara D. Neusel; Larry Smith*

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.

#### Readership

Graduate students and research mathematicians interested in commutative rings, algebras, and algebraic topology.

#### Reviews & Endorsements

[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

# Table of Contents

## Invariant Theory of Finite Groups

- Contents v6 free
- 1. Invariants, their Relatives, and Problems 110 free
- 1.1 Polynomial Invariants of Linear Groups 211
- 1.2 Coinvariants and Stable Invariants 817
- 1.3 Basic Problems in Invariant Theory 1221
- 1.4 Problems for Finite Groups 1524
- 1.5 Problems for Finite Groups over Finite Fields 2029
- 1.6 Problems for Special Representations 2332
- 1.7 What Makes Rings of Invariants Special? 2534

- 2. Algebraic Finiteness 2938
- 3. Combinatorial Finiteness 4554
- 4. Noetherian Finiteness 7786
- 5. Homological Finiteness 113122
- 5.1 The Koszul Complex 114123
- 5.2 Hilbert's Syzygy Theorem 118127
- 5.3 The Converse of Hilbert's Syzygy Theorem 120129
- 5.4 Poincaré Duality Algebras 124133
- 5.5 The Cohen–Macaulay Property 129138
- 5.6 Homological and Cohomological Dimensions 137146
- 5.7 The Gorenstein and Other Homological Properties 143152
- 5.8 Examples 147156

- 6. Modular Invariant Theory 151160
- 7. Special Classes of Invariants 185194
- 8. The Steenrod Algebra and Invariant Theory 227236
- 9. Invariant Ideals 259268
- 10. Lannes's T-Functor and Applications 283292
- 10.1 The T-Functor and Invariant Theory 284293
- 10.2 The T-Functor and Noetherian Finiteness 290299
- 10.3 Change of Rings for Components 294303
- 10.4 The T-Functor and Freeness 298307
- 10.5 The T-Functor and Complete Intersections 303312
- 10.6 Invariants of Stabilizer Subgroups 307316
- 10.7 A Last Look at the Transfer 310319

- Appendix A. Review of Commutative Algebra 315324
- References 331340
- Typography 357366
- Notation 359368
- Index 363372