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El. knyga: Representation Theory of Finite Groups: a Guidebook

  • Formatas: PDF+DRM
  • Serija: Universitext
  • Išleidimo metai: 30-Aug-2019
  • Leidėjas: Springer Nature Switzerland AG
  • Kalba: eng
  • ISBN-13: 9783030217921
Kitos knygos pagal šią temą:
Representation Theory of Finite Groups: a GuidebookDidesnis paveikslėlis
  • Formatas: PDF+DRM
  • Serija: Universitext
  • Išleidimo metai: 30-Aug-2019
  • Leidėjas: Springer Nature Switzerland AG
  • Kalba: eng
  • ISBN-13: 9783030217921
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This book provides an accessible introduction to the state of the art of representation theory of finite groups. Starting from a basic level that is summarized at the start, the book proceeds to cover topics of current research interest, including open problems and conjectures.





The central themes of the book are block theory and module theory of group representations, which are comprehensively surveyed with a full bibliography. The individual chapters cover a range of topics within the subject, from blocks with cyclic defect groups to representations of symmetric groups.





Assuming only modest background knowledge at the level of a first graduate course in algebra, this guidebook, intended for students taking first steps in the field, will also provide a reference for more experienced researchers. Although no proofs are included, end-of-chapter exercises make it suitable for student seminars.

Recenzijos

The book is intended for students and general readers. ... each of the chapters is followed by a series of exercises (83 in all), which are suitable for student seminars. (A. S. Kondrat'ev, Mathematical Reviews, November, 2020)

The book is highly recommended for researchers in representation and character theory of finite groups. (Mohammad-Reza Darafsheh, zbMATH 1446.20002, 2020)

1 The Basics
1(14)
1.1 Representation Theory
1(6)
1.2 Group Theory
7(8)
2 Blocks and Their Characters
15(22)
2.1 Blocks and Block Idempotents
17(2)
2.2 Brauer Characters
19(9)
2.3 Defect
28(5)
2.4 Brauer Correspondence
33(4)
3 Modules
37(42)
3.1 Projectives
38(7)
3.2 Vertices, Sources and the Green Correspondence
45(8)
3.3 The Module Category
53(8)
3.4 Extensions
61(5)
3.5 The Stable and Derived Categories
66(13)
4 The Local-Global Principle
79(28)
4.1 Brauer's Height-Zero Conjecture
80(3)
4.2 The McKay Conjecture
83(4)
4.3 Alperin's Weight Conjecture
87(2)
4.4 Broue's Abelian Defect Group Conjecture
89(5)
4.5 Donovan's Conjecture
94(6)
4.6 Feit's Conjecture
100(3)
4.7 Brauer's K(B)-Conjecture
103(4)
5 Blocks with Cyclic Defect Groups
107(26)
5.1 The Brauer Tree
107(8)
5.2 Brauer Tree Algebras
115(9)
5.3 Classification of Brauer Trees
124(9)
5.3.1 From All Groups to Simple Groups
125(1)
5.3.2 Alternating Groups
126(1)
5.3.3 Sporadic Groups
127(1)
5.3.4 Groups of Lie Type
128(5)
6 Blocks with Non-cyclic Defect Groups
133(20)
6.1 Klein Four Defect Groups
134(3)
6.2 Tame Defect Groups
137(9)
6.3 Nilpotent Blocks
146(5)
6.4 What Happens in General?
151(2)
7 Clifford Theory
153(14)
7.1 Representations and Normal Subgroups
154(4)
7.2 Group-Graded Algebras
158(2)
7.3 Extensions of Representations
160(2)
7.4 Clifford Theory of Blocks
162(5)
8 Representations of Symmetric Groups
167(46)
8.1 The Combinatorics of the Character Table
168(6)
8.2 Specht Modules
174(12)
8.3 Blocks and Decomposition Numbers of Symmetric Groups
186(14)
8.4 The Double Cover of S
200(13)
9 Representations of Groups of Lie Type
213(44)
9.1 Defining-Characteristic Representations
214(11)
9.2 Unipotent Classes and Characters
225(11)
9.3 Unipotent Blocks
236(11)
9.3.1 Distribution of Unipotent Characters into Blocks
237(2)
9.3.2 Basic Sets
239(3)
9.3.3 Unitriangularity of the Decomposition Matrix
242(1)
9.3.4 Reduction Modulo p of Cuspidal Characters
243(1)
9.3.5 Known Decomposition Matrices
244(3)
9.4 General Blocks
247(10)
References 257(22)
Index of Names 279(6)
Index 285
David A. Craven is a Royal Society Research Fellow and Senior Birmingham Fellow at the University of Birmingham. He has worked on a variety of areas within representations of groups, from the local-global conjectures to representations of symmetric groups, and groups of Lie type.
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