Modular representation theory of finite groups

183 Pages·2013·0.976 MB·English

by  P Schneider

Modular Representation Theory of Finite Groups
Peter Schneider
Modular
Representation
Theory of Finite
Groups
PeterSchneider
DepartmentofMathematics
UniversityofMünster
Münster
Germany
ISBN978-1-4471-4831-9 ISBN978-1-4471-4832-6(eBook)
DOI10.1007/978-1-4471-4832-6
SpringerDordrechtHeidelbergNewYorkLondon
LibraryofCongressControlNumber:2012954001
MathematicsSubjectClassification: 20C20,20C05
©Springer-VerlagLondon2013
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Preface
The nature of the representation theory of a finite group G in (finite-dimensional)
vector spaces over some field k depends very much on the relation between the
order |G| of the group G and the characteristic char(k) of the field k. If char(k)
doesnotdivide |G| thenallrepresentationsaresemisimple,i.e. aredirectsumsof
irreduciblerepresentations.Thereasonforthisisthesemisimplicityofthegroupal-
gebrak[G]inthissituation.BythemodularrepresentationtheoryofGonemeans,
ontheotherhand,thecasewherechar(k)isadivisorof|G|(sothat,inparticular,
char(k)mustbeaprimenumber).Thegroupalgebrak[G]nowmaybefarfrombe-
ingsemisimple.Intheextremecase,forexample,where|G|isapowerofchar(k),
itisalocalring;thereisthenasingleirreduciblerepresentation,whichisthetrivial
one,whereasthestructureofageneralrepresentationwillstillbeverycomplicated.
Asaconsequenceawholerangeofadditionaltoolshavetobedevelopedandused
in the course of the investigation. To mention some, there is the systematic use of
Grothendieck groups (Chap. 2) as well as Green’s direct analysis of indecompos-
ablerepresentations(Chap.4).Therealsoisthestrategyofwritingthecategoryof
allk[G]-modulesasthedirectproductofcertainsubcategories,theso-calledblocks
ofG,byusingtheactionoftheprimitiveidempotentsinthecenterofk[G].Brauer’s
approachthenestablishescorrespondencesbetweentheblocksof Gandblocksof
certainsubgroupsofG(Chap.5),thephilosophybeingthatoneistherebyreduced
toasimplersituation.Thisallowsus,inparticular,tomeasurehownonsemisimple
acategoryablockisbythesizeandstructureofitsso-calleddefectgroup.Begin-
ninginSect.4.4alltheseconceptsaremadeexplicitfortheexampleofthegroup
G=SL (F ).
2 p
The present book is to be thought of as an introduction to the major tools and
strategiesofmodularrepresentationtheory.Itscontentwastaughtduringacourse
lasting the full academic year 2010/2011 at Münster. Some basic algebra together
with the semisimple case were assumed to be known, although all facts to be
used are restated (without proofs) in the text. Otherwise the book is entirely self-
contained.Thereferences[1–10]provideacompletelistofthesourcesIhavedrawn
upon. Of course, there already exist several textbooks on the subject. The older
oneslike[5]and[6]arewritteninamostlygrouptheoreticlanguage.Thebeautiful
v
vi Preface
book [1] develops the theory entirely from the module theoretic point of view but
leavesoutcompletelythecomparisonwithgrouptheoreticconcepts.Forexample,
theconceptofdefectgroupscanbeintroducedeitherpurelygrouptheoreticallyor
purelymoduletheoretically.Tomyknowledgeallexistingbooksessentiallyrestrict
themselvestoadiscussionofoneoftheseapproachesonly.Althoughmypresenta-
tionisstronglybiasedtowardsthemoduletheoreticpointofview,Imakeanattempt
tostrikeacertainbalancebyalsoshowingthereadertheotheraspect.Inparticular,
inthecaseofdefectgroupsadetailedproofoftheequivalenceofthetwoapproaches
willbegiven.
This book is not addressed to experts. It does not discuss any very advanced
aspectsnoranyspecializedresultsofthetheory.Theaimistofamiliarizestudents
at the masters level with the basic results, tools, and techniques of a beautiful and
important algebraic theory, hopefully enabling them to subsequently pursue their
ownmorespecializedproblems.
IwishtothankT.SchmidtforcarefullyreadingafirstdraftandI.Reckermann
andG.Dierkesfortheirexcellenttypesettingofthemanuscript.
Münster,Germany PeterSchneider
Contents
1 PrerequisitesinModuleTheory . . . . . . . . . . . . . . . . . . . . . 1
1.1 ChainConditionsandMore . . . . . . . . . . . . . . . . . . . . . 1
1.2 Radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 I-AdicCompleteness . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 UniqueDecomposition . . . . . . . . . . . . . . . . . . . . . . . 9
1.5 IdempotentsandBlocks . . . . . . . . . . . . . . . . . . . . . . . 15
1.6 ProjectiveModules . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.7 GrothendieckGroups . . . . . . . . . . . . . . . . . . . . . . . . 34
2 TheCartan–BrauerTriangle . . . . . . . . . . . . . . . . . . . . . . 43
2.1 TheSetting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.2 TheTriangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.3 TheRingStructureofR (G),andInduction . . . . . . . . . . . . 54
F
2.4 TheBurnsideRing . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.5 CliffordTheory . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
2.6 Brauer’sInductionTheorem . . . . . . . . . . . . . . . . . . . . . 71
2.7 SplittingFields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
2.8 PropertiesoftheCartan–BrauerTriangle . . . . . . . . . . . . . . 78
3 TheBrauerCharacter . . . . . . . . . . . . . . . . . . . . . . . . . . 87
3.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
3.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4 Green’sTheoryofIndecomposableModules . . . . . . . . . . . . . . 97
4.1 RelativelyProjectiveModules . . . . . . . . . . . . . . . . . . . . 97
4.2 VerticesandSources . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.3 TheGreenCorrespondence . . . . . . . . . . . . . . . . . . . . . 110
4.4 AnExample:TheGroupSL (F ) . . . . . . . . . . . . . . . . . . 119
2 p
4.5 Green’sIndecomposabilityTheorem . . . . . . . . . . . . . . . . 140
5 Blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.1 BlocksandSimpleModules . . . . . . . . . . . . . . . . . . . . . 147
vii
viii Contents
5.2 CentralCharacters . . . . . . . . . . . . . . . . . . . . . . . . . . 151
5.3 DefectGroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
5.4 TheBrauerCorrespondence . . . . . . . . . . . . . . . . . . . . . 159
5.5 BrauerHomomorphisms . . . . . . . . . . . . . . . . . . . . . . . 165
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
Chapter 1
Prerequisites in Module Theory
Let R be an arbitrary (not necessarily commutative) ring (with unit). By an
R-modulewewillalwaysmeanaleftR-module.Allringhomomorphismsrespect
theunitelement,butasubringmayhaveadifferentunitelement.
1.1 ChainConditionsand More
ForanR-moduleM wehavethenotionsofbeing
finitelygenerated,artinian,noetherian,simple,andsemisimple.
TheringRiscalledleftartinian,resp.leftnoetherian,resp.semisimple,ifithasthis
propertyasaleftmoduleoveritself.
Proposition1.1.1
i. The R-module M is noetherian if and only if any submodule of M is finitely
generated.
ii. LetL⊆M beasubmodule;thenM isartinian,resp.noetherian,ifandonlyif
LandM/Lareartinian,resp.noetherian.
iii. IfRisleftartinian,resp.leftnoetherian,theneveryfinitelygeneratedR-module
M isartinian,resp.noetherian.
iv. If R is left noetherian then an R-module M is noetherian if and only if it is
finitelygenerated.
Proposition1.1.2(Jordan–Hölder) ForanyR-moduleM thefollowingconditions
areequivalent:
i. M isartinianandnoetherian;
ii. M has a composition series {0}=M ⊆M ⊆···⊆M =M such that all
0 1 n
Mi/Mi−1 aresimpleR-modules.
P.Schneider,ModularRepresentationTheoryofFiniteGroups, 1
DOI10.1007/978-1-4471-4832-6_1,©Springer-VerlagLondon2013
2 1 PrerequisitesinModuleTheory
In this case two composition series {0}=M ⊆M ⊆···⊆M =M and {0}=
0 1 n
L0 ⊆L1 ⊆···⊆Lm =M satisfy n=m and Li/Li−1 ∼=Mσ(i)/Mσ(i)−1, for any
1≤i≤m,whereσ isanappropriatepermutationof{1,...,n}.
AnR-moduleM whichsatisfiestheconditionsofProposition1.1.2iscalledof
finitelengthandtheintegerl(M):=nisitslength.Let
Rˆ :=setofallisomorphismclassesofsimpleR-modules.
Forτ ∈Rˆ andanR-moduleM theτ-isotypiccomponentofM is
M(τ):=sumofallsimplesubmodulesofM inτ.
Lemma1.1.3 ForanyR-modulehomomorphismf :L−→M wehavef(L(τ))⊆
M(τ).
Proposition1.1.4
i. ForanyR-moduleM thefollowingconditionsareequivalent:
a. M issemisimple,i.e.isomorphictoadirectsumofsimpleR-modules;
b. M isthesumofitssimplesubmodules;
c. everysubmoduleofM hasacomplement.
ii. Submodulesandfactormodulesofsemisimplemodulesaresemisimple.
iii. IfRissemisimplethenanyR-moduleissemisimple.
iv. Anyτ-isotypiccomponentM(τ)ofanyR-m(cid:2)oduleM issemisimple.
v. IftheR-moduleM issemisim(cid:3)plethenM= τ∈RˆM(τ).
vi. IfRissemisimplethenR= τ∈RˆR(τ)asrings.
Lemma 1.1.5 Any R-module M contains a unique maximal submodule which is
semisimple(andwhichiscalledthesoclesoc(M)ofM).
(cid:4)
Proof We define soc(M):= τ∈RˆM(τ). By Proposition 1.1.4.i the submodule
soc(M) is sem(cid:4)isimple. On the other hand if L⊆M is any semisimple submod-
ule then L= L(τ) by Proposition 1.1.4.v. But L(τ)⊆M(τ) by Lemma 1.1.3
τ
andhenceL⊆soc(M). (cid:2)
Definition AnR-moduleM iscalleddecomposableifthereexistnonzerosubmod-
ulesM ,M ⊆M suchthatM=M ⊕M ;correspondingly,M iscalledindecom-
1 2 1 2
posableifitisnonzeroandnotdecomposable.
Lemma 1.1.6 If M is artinian or noetherian then M is the direct sum of finitely
manyindecomposablesubmodules.
Proof We may assume that M (cid:8)={0}. Step 1: We claim that M has a nonzero in-
decomposable direct summand N. If M is artinian take a minimal element N of