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. 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. Preface......Page 6 Contents......Page 8 1.1 Representation Theory......Page 10 1.2 Group Theory......Page 16 Exercises......Page 21 2 Blocks and Their Characters......Page 24 2.1 Blocks and Block Idempotents......Page 26 2.2 Brauer Characters......Page 28 2.3 Defect......Page 37 2.4 Brauer Correspondence......Page 42 Exercises......Page 43 3 Modules......Page 46 3.1 Projectives......Page 47 3.2 Vertices, Sources and the Green Correspondence......Page 54 3.3 The Module Category......Page 62 3.4 Extensions......Page 70 3.5 The Stable and Derived Categories......Page 75 Exercises......Page 84 4 The Local-Global Principle......Page 87 4.1 Brauer's Height-Zero Conjecture......Page 88 4.2 The McKay Conjecture......Page 91 4.3 Alperin's Weight Conjecture......Page 95 4.4 Broué's Abelian Defect Group Conjecture......Page 97 4.5 Donovan's Conjecture......Page 102 4.6 Feit's Conjecture......Page 108 4.7 Brauer's k(B)-Conjecture......Page 111 Exercises......Page 113 5.1 The Brauer Tree......Page 114 5.2 Brauer Tree Algebras......Page 122 5.3 Classification of Brauer Trees......Page 131 5.3.1 From All Groups to Simple Groups......Page 132 5.3.2 Alternating Groups......Page 133 5.3.3 Sporadic Groups......Page 134 5.3.4 Groups of Lie Type......Page 135 Exercises......Page 137 6 Blocks with Non-cyclic Defect Groups......Page 139 6.1 Klein Four Defect Groups......Page 140 6.2 Tame Defect Groups......Page 143 6.3 Nilpotent Blocks......Page 152 Exercises......Page 157 7 Clifford Theory......Page 159 7.1 Representations and Normal Subgroups......Page 160 7.2 Group-Graded Algebras......Page 164 7.3 Extensions of Representations......Page 166 7.4 Clifford Theory of Blocks......Page 168 Exercises......Page 172 8 Representations of Symmetric Groups......Page 173 8.1 The Combinatorics of the Character Table......Page 174 8.2 Specht Modules......Page 180 8.3 Blocks and Decomposition Numbers of Symmetric Groups......Page 192 8.4 The Double Cover of Sn......Page 206 Exercises......Page 216 9 Representations of Groups of Lie Type......Page 219 9.1 Defining-Characteristic Representations......Page 220 9.2 Unipotent Classes and Characters......Page 231 9.3 Unipotent Blocks......Page 242 9.3.1 Distribution of Unipotent Characters into Blocks......Page 243 9.3.2 Basic Sets......Page 245 9.3.3 Unitriangularity of the Decomposition Matrix......Page 248 9.3.4 Reduction Modulo p of Cuspidal Characters......Page 249 9.3.5 Known Decomposition Matrices......Page 250 9.4 General Blocks......Page 253 Exercises......Page 259 References......Page 262 Index of Names......Page 283 Index......Page 288