Representation Theory of Finite Groups: A Guidebook PDF
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Universitext David A. Craven Representation Theory of Finite Groups: a Guidebook Universitext Universitext Serieseditors SheldonAxler SanFranciscoStateUniversity,SanFrancisco,USA CarlesCasacuberta UniversitatdeBarcelona,Barcelona,Spain AngusMacIntyre QueenMaryUniversityofLondon,London,UnitedKingdom KennethRibet UniversityofCalifornia,Berkeley,USA ClaudeSabbah Écolepolytechnique,CNRS,UniversitéParis-Saclay,Palaiseau,France EndreSüli UniversityofOxford,Oxford,UnitedKingdom WojborA.Woyczyn´ski CaseWesternReserveUniversity,Cleveland,USA Universitext is a series of textbooksthat presents material from a wide variety of mathematicaldisciplinesatmaster’slevelandbeyond.Thebooks,oftenwellclass- testedbytheirauthor,mayhaveaninformal,personalevenexperimentalapproach to their subject matter. Some of the most successful and established books in the series have evolved through several editions, always following the evolution of teachingcurricula,intoverypolishedtexts. 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Craven Representation Theory of Finite Groups: a Guidebook 123 DavidA.Craven SchoolofMathematics UniversityofBirmingham UnitedKingdom ISSN0172-5939 ISSN2191-6675 (electronic) Universitext ISBN978-3-030-21791-4 ISBN978-3-030-21792-1 (eBook) https://doi.org/10.1007/978-3-030-21792-1 MathematicsSubjectClassification(2010):20C20,20C15,20C30,20C33,20C08 ©SpringerNatureSwitzerlandAG2019 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. 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Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface The representation theory of finite groups has, at its core, a collection of open problems.Takentogether,theyarecalled‘local-globalconjectures’,although‘local- global rough guesses’ might be more appropriate. The missing pieces of the representation-theoreticpuzzle lie right in the centre, leaving us with corners and edgestofittogether. Many of the conjectures are numerical in nature, telling us that there is some exquisiteunderlyingstructuraltheory,butwecannotyetperceiveit.Thenumerical results are attractive in their own right, butit is the potentialhiddenstructure that makesthe subjectso captivating.Oneof themainpurposesofthistextisto make thisfieldaccessibletoanyonewithabasicunderstandingofrepresentationtheory, suchasthatprovidedbyanundergraduatecourseinit,andsomegeneralknowledge ofalgebra. Thelocal-globalconjecturesmightformthecentrepieceoffinitegrouprepresen- tationtheory,butthereismuchelseonthetablearoundthem:thetheoryofblocks withacyclicdefectgroup,therepresentationtheoryofthesymmetricgroups,and ofthegroupsofLietype,thestructureoftameblocks,andmuchmorebesides. The aim of this book is to give the reader a guided tour of these areas. Like a guide book, we seek to uncover the highlights of the field; proofs are more or less completely excised from this text, although the intrepid reader, using this as a springboard for further study, will be delighted to find that there are literally hundredsofreferencestoconsultformoredetails. Ofcourse,likeanyguidebook,itstartstobecomeoutofdateonceitiswritten. Thiswillsetinaspicthestate ofthetheoryattheearlypartof2019,butIbelieve thatmuchofthistheorywillremainrelevantfordecadestocome. Thefirstchapterquicklysummarizeswhatisneededaboutrepresentationtheory, andthesecondandthirdchaptersgothroughthebasicbuildingblocksofthetheory, namelythecharactertheoryandthemoduletheory. Theremainingchaptersaremoreorlessindependentofoneanother,thoughthere willbecrossreferencing.Thisshouldallowreaderstodipinandoutofthebookas andwhentheywanttoknowaboutaspecificaspectofthetheory. v vi Preface The local-global conjectures occupy Chap.4, the well-developed theory of blocks with cyclic defect groups following it in Chap.5. Chapter 6 attempts to givesomeroughideasaboutthestructureofblockswithnon-cyclicdefectgroups, althoughverylittleisknowninthiscase.ThenextchapterdealswithCliffordtheory, therelationshipbetweenmodulesforagroupandanormalsubgroup.Thelasttwo chapters give some of the representation theory of symmetric groups and of the groupsofLietype. Ialsohavetothankpeopleforhelpingfindsomeofthemanyerrorsthatoccurred in previous versions of this text. In particular, I thank David Benson, Norman McGregor, John Murray, Gabriel Navarro, Jeremy Rickard, Geoffrey Robinson, AdamThomas,MarkWildon,andespeciallyGunterMallefortheirsuggestions. Birmingham,UK DavidA.Craven Contents 1 TheBasics..................................................................... 1 1.1 RepresentationTheory.................................................. 1 1.2 GroupTheory ........................................................... 7 2 BlocksandTheirCharacters............................................... 15 2.1 BlocksandBlockIdempotents......................................... 17 2.2 BrauerCharacters....................................................... 19 2.3 Defect.................................................................... 28 2.4 BrauerCorrespondence................................................. 33 3 Modules ....................................................................... 37 3.1 Projectives............................................................... 38 3.2 Vertices,SourcesandtheGreenCorrespondence..................... 45 3.3 TheModuleCategory................................................... 53 3.4 Extensions............................................................... 61 3.5 TheStableandDerivedCategories..................................... 66 4 TheLocal-GlobalPrinciple................................................. 79 4.1 Brauer’sHeight-ZeroConjecture....................................... 80 4.2 TheMcKayConjecture................................................. 83 4.3 Alperin’sWeightConjecture ........................................... 87 4.4 Broué’sAbelianDefectGroupConjecture ............................ 89 4.5 Donovan’sConjecture .................................................. 94 4.6 Feit’sConjecture........................................................ 100 4.7 Brauer’sk(B)-Conjecture .............................................. 103 5 BlockswithCyclicDefectGroups.......................................... 107 5.1 TheBrauerTree......................................................... 107 5.2 BrauerTreeAlgebras................................................... 115 5.3 ClassificationofBrauerTrees.......................................... 124 5.3.1 FromAllGroupstoSimpleGroups ........................... 125 5.3.2 AlternatingGroups ............................................. 126 vii viii Contents 5.3.3 SporadicGroups................................................ 127 5.3.4 GroupsofLieType............................................. 128 6 BlockswithNon-cyclicDefectGroups..................................... 133 6.1 KleinFourDefectGroups.............................................. 134 6.2 TameDefectGroups.................................................... 137 6.3 NilpotentBlocks ........................................................ 146 6.4 WhatHappensinGeneral?............................................. 151 7 CliffordTheory............................................................... 153 7.1 RepresentationsandNormalSubgroups............................... 154 7.2 Group-GradedAlgebras ................................................ 158 7.3 ExtensionsofRepresentations.......................................... 160 7.4 CliffordTheoryofBlocks.............................................. 162 8 RepresentationsofSymmetricGroups.................................... 167 8.1 TheCombinatoricsoftheCharacterTable............................. 168 8.2 SpechtModules......................................................... 174 8.3 BlocksandDecompositionNumbersofSymmetricGroups.......... 186 8.4 TheDoubleCoverofS ................................................ 200 n 9 RepresentationsofGroupsofLieType.................................... 213 9.1 Defining-CharacteristicRepresentations............................... 214 9.2 UnipotentClassesandCharacters...................................... 225 9.3 UnipotentBlocks........................................................ 236 9.3.1 DistributionofUnipotentCharactersintoBlocks............. 237 9.3.2 BasicSets ....................................................... 239 9.3.3 UnitriangularityoftheDecompositionMatrix................ 242 9.3.4 ReductionModulopofCuspidalCharacters ................. 243 9.3.5 KnownDecompositionMatrices............................... 244 9.4 GeneralBlocks.......................................................... 247 References......................................................................... 257 IndexofNames................................................................... 279 Index............................................................................... 285 Chapter 1 The Basics Thisbookaimstointroducethereadertothemodernrepresentationtheoryoffinite groups.Todosoweneedsomeofthebasicsofrepresentationtheory,whichwewill ratherskimthrough,mostlytofixnotation.Wearegoingtoassumethatthereader ismoreorlessfamiliarwiththis,andmightneedarefresheronly.Wealsoinclude aratherrusheddescriptionofthegrouptheorywe willneedforthisbook,atleast to understandthe chaptersthat are not devoted to particular groups, most notably Chaps.4and9. 1.1 Representation Theory General references for the basic representation theory of finite groups include [313, 328, 388], although any textbook on representation theory or set of lecture notes from an advanced undergraduate/beginning graduate course will do. In particular, I personally have always found [328] an unparalleled first introduction torepresentationsoffinitegroups. Let k denote a field, which will have characteristic p (p is allowed to be 0). We will almost always consider k to be algebraically closed. A representation of a group G is a homomorphismG → GL (k). This is equivalentto the notion of n a (finite-dimensional) kG-module, which we define now. The group algebra kG of a group G over a field k is the k-algebra consisting of all formal finite linear combinationsofgroupelementswithcoefficientsink (sohasabasisconsistingof thegroupelements)andwithadditionandmultiplicationgivenbytheobviouslinear actions. A right kG-module is a right module for the group algebra kG. We can simply think of this as a vector space with an action of G on it. In this book, all of our modules will be rightmodules, and finite-dimensionalunless we explicitly mentionotherwise.Ofcourse,itisjustaconventiontouserightmodules,andallour theoremswillholdtrueforleftmodules,exceptperhapswithachangeofordering ©SpringerNatureSwitzerlandAG2019 1 D.A.Craven,RepresentationTheoryofFiniteGroups:aGuidebook,Universitext, https://doi.org/10.1007/978-3-030-21792-1_1
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