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E-grāmata: Course in Finite Group Representation Theory

(University of Minnesota)
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Course in Finite Group Representation TheoryPalielināt
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This graduate-level text provides a thorough grounding in the representation theory of finite groups over fields and rings. The book provides a balanced and comprehensive account of the subject, detailing the methods needed to analyze representations that arise in many areas of mathematics. Key topics include the construction and use of character tables, the role of induction and restriction, projective and simple modules for group algebras, indecomposable representations, Brauer characters, and block theory. This classroom-tested text provides motivation through a large number of worked examples, with exercises at the end of each chapter that test the reader's knowledge, provide further examples and practice, and include results not proven in the text. Prerequisites include a graduate course in abstract algebra, and familiarity with the properties of groups, rings, field extensions, and linear algebra.

Recenzijas

'This is a well-written and motivated book, with carefully chosen topics, examples and exercises to engage the reader, making it suitable in the classroom or for self-study.' Felipe Zaldivar, MAA Reviews 'The author aims to provide a comprehensive but fastpaced grounding in results which can be applied to areas as diverse as number theory, combinatorics, topology or commutative algebra While the proofs are rigorous, the style is relatively informal and designed to showcase as many results as possible which are applicable beyond the realms of \pure representation theory.' Stuart Martin, MathSciNet

Papildus informācija

This classroom-tested graduate text provides a thorough grounding in the representation theory of finite groups over fields and rings.
Preface ix
1 Representations, Maschke's Theorem, and Semisimplicity
1(14)
1.1 Definitions and Examples
1(6)
1.2 Semisimple Representations
7(4)
1.3 Summary of
Chapter 1
11(1)
1.4 Exercises for
Chapter 1
11(4)
2 The Structure of Algebras for Which Every Module Is Semisimple
15(9)
2.1 Schur's Lemma and Wedderburn's Theorem
15(5)
2.2 Summary of
Chapter 2
20(1)
2.3 Exercises for
Chapter 2
20(4)
3 Characters
24(32)
3.1 The Character Table
24(6)
3.2 Orthogonality Relations and Bilinear Forms
30(3)
3.3 Consequences of the Orthogonality Relations
33(5)
3.4 The Number of Simple Characters
38(4)
3.5 Algebraic Integers and Divisibility of Character Degrees
42(3)
3.6 The Matrix Summands of the Complex Group Algebra
45(3)
3.7 Burnside's paqb Theorem
48(2)
3.8 Summary of
Chapter 3
50(1)
3.9 Exercises for
Chapter 3
51(5)
4 The Construction of Modules and Characters
56(26)
4.1 Cyclic Groups and Direct Products
56(3)
4.2 Lifting (or Inflating) from a Quotient Group
59(1)
4.3 Induction and Restriction
60(10)
4.4 Symmetric and Exterior Powers
70(5)
4.5 The Construction of Character Tables
75(1)
4.6 Summary of
Chapter 4
75(1)
4.7 Exercises for
Chapter 4
76(6)
5 More on Induction and Restriction: Theorems of Mackey and Clifford
82(13)
5.1 Double Cosets
82(2)
5.2 Mackey's Theorem
84(3)
5.3 Clifford's Theorem
87(3)
5.4 Summary of
Chapter 5
90(1)
5.5 Exercises for
Chapter 5
91(4)
6 Representations of p-Groups in Characteristic p and the Radical
95(20)
6.1 Cyclic p-Groups
95(3)
6.2 Simple Modules for Groups with Normal p-Subgroups
98(2)
6.3 Radicals, Socles, and the Augmentation Ideal
100(6)
6.4 Jennings's Theorem
106(2)
6.5 Summary of
Chapter 6
108(1)
6.6 Exercises for
Chapter 6
108(7)
7 Projective Modules for Finite-Dimensional Algebras
115(20)
7.1 Characterizations of Projective and Injective Modules
115(4)
7.2 Projectives by Means of Idempotents
119(3)
7.3 Projective Covers, Nakayama's Lemma, and Lifting of Idempotents
122(8)
7.4 The Cartan Matrix
130(2)
7.5 Summary of
Chapter 7
132(1)
7.6 Exercises for
Chapter 7
132(3)
8 Projective Modules for Group Algebras
135(23)
8.1 The Behavior of Projective Modules under Induction, Restriction, and Tensor Product
135(3)
8.2 Projective and Simple Modules for Direct Products of a p-Group and a p1-Group
138(2)
8.3 Projective Modules for Groups with a Normal Sylow p-Subgroup
140(5)
8.4 Projective Modules for Groups with a Normal p-Complement
145(2)
8.5 Symmetry of the Group Algebra
147(5)
8.6 Summary of
Chapter 8
152(1)
8.7 Exercises for
Chapter 8
152(6)
9 Changing the Ground Ring: Splitting Fields and the Decomposition Map
158(34)
9.1 Some Definitions
159(1)
9.2 Splitting Fields
159(4)
9.3 The Number of Simple Representations in Positive Characteristic
163(4)
9.4 Reduction Modulo p and the Decomposition Map
167(10)
9.5 The cde Triangle
177(5)
9.6 Blocks of Defect Zero
182(4)
9.7 Summary of
Chapter 9
186(1)
9.8 Exercises for
Chapter 9
187(5)
10 Brauer Characters
192(21)
10.1 The Definition of Brauer Characters
192(6)
10.2 Orthogonality Relations and Grothendieck Groups
198(7)
10.3 The cde Triangle in Terms of Brauer Characters
205(3)
10.4 Summary of
Chapter 10
208(1)
10.5 Exercises for
Chapter 10
208(5)
11 Indecomposable Modules
213(44)
11.1 Indecomposable Modules, Local Rings, and the Krull-Schmidt Theorem
214(4)
11.2 Groups with a Normal Cyclic Sylow p-Subgroup
218(1)
11.3 Relative Projectivity
219(9)
11.4 Finite Representation Type
228(4)
11.5 Infinite Representation Type and the Representations of C2 × C2
232(5)
11.6 Vertices, Sources, and Green Correspondence
237(8)
11.7 The Heller Operator
245(3)
11.8 Some Further Techniques with Indecomposable Modules
248(1)
11.9 Summary of
Chapter 11
249(1)
11.10 Exercises for
Chapter 11
250(7)
12 Blocks
257
12.1 Blocks of Rings in General
258(4)
12.2 p-Blocks of Groups
262(4)
12.3 The Defect of a Block: Module Theoretic Methods
266
Peter Webb is Professor of Mathematics at the University of Minnesota. His research interests focus on the interactions between group theory and other areas of algebra, combinatorics, and topology. In 1988, he was awarded a Whitehead Prize of the London Mathematical Society.
Biežāk uzdotie jautājumi par e-grāmatām Biežāk uzdotie jautājumi par e-grāmatām
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