Representation Theory of Finite Groups: A Guidebook

Description

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. Preface Contents 1 The Basics 1.1 Representation Theory 1.2 Group Theory Exercises 2 Blocks and Their Characters 2.1 Blocks and Block Idempotents 2.2 Brauer Characters 2.3 Defect 2.4 Brauer Correspondence Exercises 3 Modules 3.1 Projectives 3.2 Vertices, Sources and the Green Correspondence 3.3 The Module Category 3.4 Extensions 3.5 The Stable and Derived Categories Exercises 4 The Local-Global Principle 4.1 Brauer's Height-Zero Conjecture 4.2 The McKay Conjecture 4.3 Alperin's Weight Conjecture 4.4 Broué's Abelian Defect Group Conjecture 4.5 Donovan's Conjecture 4.6 Feit's Conjecture 4.7 Brauer's k(B)-Conjecture Exercises 5 Blocks with Cyclic Defect Groups 5.1 The Brauer Tree 5.2 Brauer Tree Algebras 5.3 Classification of Brauer Trees 5.3.1 From All Groups to Simple Groups 5.3.2 Alternating Groups 5.3.3 Sporadic Groups 5.3.4 Groups of Lie Type Exercises 6 Blocks with Non-cyclic Defect Groups 6.1 Klein Four Defect Groups 6.2 Tame Defect Groups 6.3 Nilpotent Blocks 6.4 What Happens in General? Exercises 7 Clifford Theory 7.1 Representations and Normal Subgroups 7.2 Group-Graded Algebras 7.3 Extensions of Representations 7.4 Clifford Theory of Blocks Exercises 8 Representations of Symmetric Groups 8.1 The Combinatorics of the Character Table 8.2 Specht Modules 8.3 Blocks and Decomposition Numbers of Symmetric Groups 8.4 The Double Cover of Sn Exercises 9 Representations of Groups of Lie Type 9.1 Defining-Characteristic Representations 9.2 Unipotent Classes and Characters 9.3 Unipotent Blocks 9.3.1 Distribution of Unipotent Characters into Blocks 9.3.2 Basic Sets 9.3.3 Unitriangularity of the Decomposition Matrix 9.3.4 Reduction Modulo p of Cuspidal Characters 9.3.5 Known Decomposition Matrices 9.4 General Blocks Exercises References Index of Names Index