Invariant theory of finite groups / Mara D. Neusel, Larry Smith.

Format:

Book

Author/Creator:

Neusel, Mara D., 1964-

Contributor:

Smith, L. (Larry), 1942-

Series:

Mathematical surveys and monographs ; no. 94.

Mathematical surveys and monographs, 0076-5376 ; v. 94

Language:

English

Subjects (All):

Finite groups.

Invariants.

Physical Description:

vii, 371 pages : illustrations ; 27 cm.

Place of Publication:

Providence, R.I. : American Mathematical Society, [2002]

Contents:

1. Invariants, their Relatives, and Problems 1

1.1 Polynomial Invariants of Linear Groups 2

1.2 Coinvariants and Stable Invariants 8

1.3 Basic Problems in Invariant Theory 12

1.4 Problems for Finite Groups 15

1.5 Problems for Finite Groups over Finite Fields 20

1.6 Problems for Special Representations 23

1.7 What Makes Rings of Invariants Special? 25

2. Algebraic Finiteness 29

2.1 Emmy Noether's Finiteness Theorem 30

2.2 The Transfer Homomorphism 33

2.3 Emmy Noether's Bound 36

2.4 Feshbach's Transfer Theorem 40

3. Combinatorial Finiteness 45

3.1 Molien's Theorem on Poincare Series 46

3.2 Poincare Series of Permutation Representations 57

3.3 The Hilbert-Serre Theorem on Poincare Series 66

3.4 Gobel's Theorem on Permutation Invariants 69

4. Noetherian Finiteness 77

4.1 Orbit Chern Classes 78

4.2 A Refinement of Orbit Chern Classes 85

4.3 Dade Bases and Systems of Parameters 99

4.4 Euler Classes and Related Constructions 103

4.5 The Degree Theorem 105

5. Homological Finiteness 113

5.1 The Koszul Complex 114

5.2 Hilbert's Syzygy Theorem 118

5.3 The Converse of Hilbert's Syzygy Theorem 120

5.4 Poincare Duality Algebras 124

5.5 The Cohen-Macaulay Property 129

5.6 Homological and Cohomological Dimensions 137

5.7 The Gorenstein and Other Homological Properties 143

6. Modular Invariant Theory 151

6.1 The Dickson Algebra 152

6.2 Transvection Groups 156

6.3 p-Groups in Characteristic p 160

6.4 The Transfer Variety 168

6.5 The Koszul Complex and Invariant Theory 173

7. Special Classes of Invariants 185

7.1 Pseudoreflections and Pseudoreflection Groups 186

7.2 Coinvariants of Pseudoreflection Groups 194

7.3 Solvable, Nilpotent and Alternating Groups 203

7.4 GL(2, F[subscript p]) and Some of Its Subgroups 212

7.5 Integer Representations of Finite Groups 221

8. The Steenrod Algebra and Invariant Theory 227

8.1 The Steenrod Operations 228

8.2 The Steenrod Algebra 231

8.3 The Hopf Algebra Structure of the Steenrod Algebra 236

8.4 The Inverse Invariant Theory Problem 241

8.5 The Landweber-Stong Conjecture 246

8.6 The Steenrod Algebra and the Dickson Algebra 255

9. Invariant Ideals 259

9.1 Invariant Ideals and the J-Construction 260

9.2 The Invariant Prime Ideal Spectrum 266

9.3 Applications to the Transfer 275

9.4 Applications to Homological Properties 278

10. Lannes's T-Functor and Applications 283

10.1 The T-Functor and Invariant Theory 284

10.2 The T-Functor and Noetherian Finiteness 290

10.3 Change of Rings for Components 294

10.4 The T-Functor and Freeness 298

10.5 The T-Functor and Complete Intersections 303

10.6 Invariants of Stabilizer Subgroups 307

10.7 A Last Look at the Transfer 310

Appendix A. Review of Commutative Algebra 315

A.1 Gradings 315

A.2 Primary Decompositions and Integral Extensions 320

A.3 Noetherian Algebras 323

A.4 Graded Algebras and Modules 327.

Notes:

Includes bibliographical references (pages 331-355) and index.

ISBN:

0821829165

OCLC:

48098583