具体描述
This book presents the characteristic zero invariant theory of finite groups acting linearly on polynomial algebras. The author assumes basic knowledge of groups and rings, and introduces more advanced methods from commutative algebra along the way. The theory is illustrated by numerous examples and applications to physics, engineering, numerical analysis, combinatorics, coding theory, and graph theory. A wide selection of exercises and suggestions for further reading makes the book appropriate for an advanced undergraduate or first-year graduate level course.
《Invariant Theory》是一部深入探讨数学结构与对称性关系的精彩著作，其核心在于揭示变量之间保持不变性的规律。书中详细阐述了群作用下的对称元素及其消除过程，系统分析了不同类型无穷对象和极小子群之间的关联。这本书通过严谨的数学推导与丰富的案例研究，帮助读者理解抽象的数学概念如何在具体问题中得到应用。内容涵盖了从基础定义到高级推论的全面探索，适合对数学理论有深入兴趣的学习者。作者采用清晰逻辑的结构，使复杂的理论步骤易于理解，同时结合历史背景和当前研究动态，为读者提供全面的视角。书中不仅重视数学证明的严密性，还注重展示思想深度，体现了对这一主题的系统关注与执着。无论是初学者还是专业研究者，都能从中获得有价值的见解和灵感。这本书以其独特的学术深度和易读性，成为理解该领域重要内容的重要参考。 通过全篇章的深入剖析，读者将能够掌握Invariant Theory这一重要的数学分支，其内在的美妙与广泛应用。作者对理论框架的清晰刻画，以及对实际问题的诠释，使书籍成为连接基础概念和高级思考的重要桥梁。同时，书中大量引用经典文献和当前研究进展，不仅丰富了内容，还为读者提供了未来探索的方向。这本书不仅传递知识，更激发了对数学抽象与创造力的深刻感悟。它适合那些希望深入理解数学结构、探索对称性规律的人士，无论是在学术研究还是教学实践中，都具有重要的参考价值。这种内容设计无一不体现出作者对专业领域的热情与细致，为读者打开了更广阔的知识世界。 该书的另一个亮点在于其交织了历史、方法和应用的多维度内容，帮助读者从不同角度认识Invariant Theory的内涵。作者对每个概念的精确定义与严谨推导，不仅强化了理解能力，还培养了批判性思维。书中许多图示与实例分析，使复杂的数学关系更加直观，便于学习者快速掌握关键点。此外，作者还注重对现代发展方向的探讨，如如何将传统理论应用于当前技术问题，这为读者提供了更广阔的思考空间。总体而言，这是一部内容扎实、结构清晰、深度独到的学术著作，非常适合有志深入研究该领域的人士。通过阅读，读者不仅能掌握专业知识，还能提升自身的数学分析与逻辑思维能力，为未来的学习和工作打下坚实基础。
作者简介
Mara D. Neusel: Texas Tech University, Lubbock, TX
目录信息
Cover 1
Title 2
Copyright 3
Contents 6
Introduction 10
Part 1. Recollections 16
Chapter 1. Linear Representations of Finite Groups 18
§1.1. Groups 18
§1.2. Homomorphisms of Groups 23
§1.3. Linear Representation of Groups 27
§1.4. Exercises 33
Chapter 2. Rings and Algebras 36
§2.1. Rings 36
§2.2. Homomorphisms of Rings, Algebras 40
§2.3. Ideals 42
§2.4. Exercises 49
Part 2. Introduction and Göbel's Bound 52
Chapter 3. Rings of Polynomial Invariants 54
§3.1. Linear Group Actions 54
§3.2. Rings of Invariants 62
S3.3. Exercises 69
Chapter 4. Permutation Representations 72
§4.1. Permutation Representations 72
§4.2. Newton, Waring, and Gauss 76
§4.3. Göbel's Bound 84
§4.4. Exercises 96
Application: Decay of a Spinless Particle 100
Application: Counting Weighted Graphs 104
Part 3. The First Fundamental Theorem of Invariant Theory and Noether's Bound 106
Chapter 5. Construction of Invariants 108
§5.1. Orbit Chern Classes 108
§5.2. The Transfer 112
§5.3. New Invariants from Old Ones 118
§5.4. Exercises 122
Chapter 6. Noether's Bound 126
§6.1. The Noether Map 126
§6.2. Polarizations 132
§6.3. The First Fundamental Theorem of Invariant Theory for Σ[sub(d)] 136
§6.4. Noether's Bound 139
§6.5. Exercises 144
Chapter 7. Some Families of Invariants 146
§7.1. Invariants of Representations of Cyclic Groups 146
§7.2. Pseudoreflection Groups 150
§7.3. Vector Invariants 157
§7.4. Exercises 160
Application: Production of Fibre Composites 164
Application: Gaussian Quadrature 168
Part 4. Noether's Theorems 168
Chapter 8. Modules 174
§8.1. Modules and Module Homomorphisms 174
§8.2. Operations on Modules 179
§8.3. Direct Sums, Free Modules, and Finite Generatedness 181
§8.4. Maschke's Theorem and Schur's Lemma 185
§8.5. Exercises 191
Chapter 9. Integral Dependence and the Krull Relations 194
§9.1. Integral Dependence 194
§9.2. Integral Extensions 199
§9.3. The Krull Relations 202
§9.4. Exercises 207
Chapter 10. Noether's Theorems 210
§10.1. Finite Generatedness 210
§10.2. The Krull Dimension 217
§10.3. Examples 221
§10.4. Exercises 226
Application: Self-Dual Codes 228
Part 5. Advanced Counting Methods and the Shephard-Todd-Chevalley Theorem 232
Chapter 11. Poincaré Series 234
§11.1. Poincaré Series 234
§11.2. Molien's Theorem 242
§11.3. Exercises 251
Chapter 12. Systems of Parameters 254
§12.1. System of Parameters 254
§12.2. Hilbert's Nullstellensatz 258
§12.3. Primary and Secondary Generators 261
§12.4. Dade's Basis 267
§12.5. The Degree Theorem 270
§12.6. Exercises 274
Chapter 13. Pseudoreflection Representations 276
§13.1. Pseudoreflections 276
§13.2. The Shephard-Todd-Chevalley Theorem, Part I 279
§13.3. The Shephard-Todd-Chevalley Theorem, Part II 284
§13.4. Exercises 290
Application: Counting Partitions 292
Appendix A. Rational Invariants 296
§A.1. Algebraic Field Extensions 296
§A.2. Splitting Fields and Normal Extensions 302
§A.3. Galois Extensions 304
§A.4. Examples 307
§A.5. Exercises 311
Suggestions for Further Reading 312
Notation Index 314
Subject Index 316
A 316
B 316
C 316
D 317
E 317
F 318
G 318
H 319
I 319
J 319
K 319
L 320
M 320
N 320
O 321
P 321
Q 321
R 321
S 323
T 323
U 323
V 323
W 323
X 323
Y 323
Z 323
Back Cover 326
· · · · · · (收起)
Title 2
Copyright 3
Contents 6
Introduction 10
Part 1. Recollections 16
Chapter 1. Linear Representations of Finite Groups 18
§1.1. Groups 18
§1.2. Homomorphisms of Groups 23
§1.3. Linear Representation of Groups 27
§1.4. Exercises 33
Chapter 2. Rings and Algebras 36
§2.1. Rings 36
§2.2. Homomorphisms of Rings, Algebras 40
§2.3. Ideals 42
§2.4. Exercises 49
Part 2. Introduction and Göbel's Bound 52
Chapter 3. Rings of Polynomial Invariants 54
§3.1. Linear Group Actions 54
§3.2. Rings of Invariants 62
S3.3. Exercises 69
Chapter 4. Permutation Representations 72
§4.1. Permutation Representations 72
§4.2. Newton, Waring, and Gauss 76
§4.3. Göbel's Bound 84
§4.4. Exercises 96
Application: Decay of a Spinless Particle 100
Application: Counting Weighted Graphs 104
Part 3. The First Fundamental Theorem of Invariant Theory and Noether's Bound 106
Chapter 5. Construction of Invariants 108
§5.1. Orbit Chern Classes 108
§5.2. The Transfer 112
§5.3. New Invariants from Old Ones 118
§5.4. Exercises 122
Chapter 6. Noether's Bound 126
§6.1. The Noether Map 126
§6.2. Polarizations 132
§6.3. The First Fundamental Theorem of Invariant Theory for Σ[sub(d)] 136
§6.4. Noether's Bound 139
§6.5. Exercises 144
Chapter 7. Some Families of Invariants 146
§7.1. Invariants of Representations of Cyclic Groups 146
§7.2. Pseudoreflection Groups 150
§7.3. Vector Invariants 157
§7.4. Exercises 160
Application: Production of Fibre Composites 164
Application: Gaussian Quadrature 168
Part 4. Noether's Theorems 168
Chapter 8. Modules 174
§8.1. Modules and Module Homomorphisms 174
§8.2. Operations on Modules 179
§8.3. Direct Sums, Free Modules, and Finite Generatedness 181
§8.4. Maschke's Theorem and Schur's Lemma 185
§8.5. Exercises 191
Chapter 9. Integral Dependence and the Krull Relations 194
§9.1. Integral Dependence 194
§9.2. Integral Extensions 199
§9.3. The Krull Relations 202
§9.4. Exercises 207
Chapter 10. Noether's Theorems 210
§10.1. Finite Generatedness 210
§10.2. The Krull Dimension 217
§10.3. Examples 221
§10.4. Exercises 226
Application: Self-Dual Codes 228
Part 5. Advanced Counting Methods and the Shephard-Todd-Chevalley Theorem 232
Chapter 11. Poincaré Series 234
§11.1. Poincaré Series 234
§11.2. Molien's Theorem 242
§11.3. Exercises 251
Chapter 12. Systems of Parameters 254
§12.1. System of Parameters 254
§12.2. Hilbert's Nullstellensatz 258
§12.3. Primary and Secondary Generators 261
§12.4. Dade's Basis 267
§12.5. The Degree Theorem 270
§12.6. Exercises 274
Chapter 13. Pseudoreflection Representations 276
§13.1. Pseudoreflections 276
§13.2. The Shephard-Todd-Chevalley Theorem, Part I 279
§13.3. The Shephard-Todd-Chevalley Theorem, Part II 284
§13.4. Exercises 290
Application: Counting Partitions 292
Appendix A. Rational Invariants 296
§A.1. Algebraic Field Extensions 296
§A.2. Splitting Fields and Normal Extensions 302
§A.3. Galois Extensions 304
§A.4. Examples 307
§A.5. Exercises 311
Suggestions for Further Reading 312
Notation Index 314
Subject Index 316
A 316
B 316
C 316
D 317
E 317
F 318
G 318
H 319
I 319
J 319
K 319
L 320
M 320
N 320
O 321
P 321
Q 321
R 321
S 323
T 323
U 323
V 323
W 323
X 323
Y 323
Z 323
Back Cover 326
· · · · · · (收起)
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