Representation Theory of Finite Group Extensions: Clifford Theory, Mackey Obstruction, and the Orbit Method

347 Pages·2022·4.673 MB·English

by  Tullio Ceccherini-Silberstein, Fabio Scarabotti, Filippo Tolli

Springer Monographs in Mathematics
Tullio Ceccherini-Silberstein
Fabio Scarabotti
Filippo Tolli
Representation
Theory
of Finite Group
Extensions
Clifford Theory, Mackey Obstruction,
and the Orbit Method
Springer Monographs in Mathematics
Editors-in-Chief
MinhyongKim,SchoolofMathematics,KoreaInstituteforAdvancedStudy,Seoul,
SouthKorea
InternationalCentreforMathematicalSciences,Edinburgh,UK
KatrinWendland,SchoolofMathematics,TrinityCollegeDublin,Dublin,Ireland
SeriesEditors
SheldonAxler,DepartmentofMathematics,SanFranciscoStateUniversity,
SanFrancisco,CA,USA
MarkBraverman,DepartmentofMathematics,PrincetonUniversity,Princeton,
NY,USA
Maria Chudnovsky, Department of Mathematics, Princeton University, Princeton,
NY,USA
TadahisaFunaki,DepartmentofMathematics,UniversityofTokyo,Tokyo,Japan
IsabelleGallagher,DépartementdeMathématiquesetApplications,EcoleNormale
Supérieure,Paris,France
SinanGüntürk,CourantInstituteofMathematicalSciences,NewYorkUniversity,
NewYork,NY,USA
ClaudeLeBris,CERMICS,EcoledesPontsParisTech,MarnelaVallée,France
Pascal Massart, Département de Mathématiques, Université de Paris-Sud, Orsay,
France
AlbertoA.Pinto,DepartmentofMathematics,UniversityofPorto,Porto,Portugal
GabriellaPinzari,DepartmentofMathematics,UniversityofPadova,Padova,Italy
Ken Ribet, Department of Mathematics, University of California, Berkeley, CA,
USA
RenéSchilling,InstituteforMathematicalStochastics,TechnicalUniversity
Dresden,Dresden,Germany
Panagiotis Souganidis, Department of Mathematics, University of Chicago,
Chicago,IL,USA
EndreSüli,MathematicalInstitute,UniversityofOxford,Oxford,UK
ShmuelWeinberger,DepartmentofMathematics,UniversityofChicago,Chicago,
IL,USA
BorisZilber,MathematicalInstitute,UniversityofOxford,Oxford,UK
Thisseriespublishesadvancedmonographsgivingwell-writtenpresentationsofthe
“state-of-the-art”infieldsofmathematicalresearchthathaveacquiredthematurity
neededforsuchatreatment.Theyaresufficientlyself-containedtobeaccessibleto
morethanjusttheintimatespecialistsofthesubject,andsufficientlycomprehensive
toremainvaluablereferencesformanyyears.Besidesthecurrentstateofknowledge
initsfield,anSMMvolumeshouldideallydescribeitsrelevancetoandinteraction
with neighbouringfields of mathematics, and give pointers to future directionsof
research.
Tullio Ceccherini-Silberstein (cid:129) Fabio Scarabotti (cid:129)
Filippo Tolli
Representation Theory of
Finite Group Extensions
Clifford Theory, Mackey Obstruction, and the
Orbit Method
TullioCeccherini-Silberstein FabioScarabotti
DipartimentodiIngegneria DipartimentoSBAI
UniversitàdegliStudidelSannio SapienzaUniversityofRome
Benevento,Italy Roma,Italy
FilippoTolli
DipartimentodiMatematicaeFisica
RomaTreUniversity
Roma,Italy
ISSN1439-7382 ISSN2196-9922 (electronic)
SpringerMonographsinMathematics
ISBN978-3-031-13872-0 ISBN978-3-031-13873-7 (eBook)
https://doi.org/10.1007/978-3-031-13873-7
©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerland
AG2022
Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether
thewhole orpart ofthematerial isconcerned, specifically therights oftranslation, reprinting, reuse
ofillustrations, recitation, broadcasting, reproductiononmicrofilmsorinanyotherphysicalway,and
transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar
ordissimilarmethodologynowknownorhereafterdeveloped.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication
doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant
protectivelawsandregulationsandthereforefreeforgeneraluse.
Thepublisher,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook
arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor
theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany
errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional
claimsinpublishedmapsandinstitutionalaffiliations.
ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG
Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland
To thememoryof mybeloved“aunties”zia
CorneliaTrevisaniandziaMarcellaTreves,
inprimis,andAdrianaMondolfi,ziaAlma
EnriquesGhiron,Augustaandzia Gina
Castelnuovo,suorCesarina,Césarine
Poiget,ziaEdaBerger Vidale,ElisaFrontali,
FiorenzaAlmagiàMaranelli,Giacometta
Limentani,zieGiovannaeReginaFranchetti,
ziaGiulianaTreves Artom,IdaSacchetti,zia
LauraZelasio,Leah Loewenwirth Reuveni,
Liviae SiddaGarofolini,MariaDanelon,
NellaMortara,Nella Volterra,Paola
Franchetti,tataRitaAlverà,ziaSilviaTreves
Levi Vidale,Teresa Tagliaferri,MlleYvonne
Bâtard,and,lastbutnotleast,AnnaPalaggi
To Cristinaand Virginia
To mymom,mysisterRossella,mywife
Stefania,andmy nieceValentina
Preface
GiventwogroupsN andH,theproblemofclassifyingthegroupsGsuchthatwe
haveashortexactsequence
1−→N −→ι G−π→H −→1
is called the extension problem. This short exact sequence (and, by abuse of
language, the group G) is called an extension of H by N. In the framework of
representationtheory,anaturalquestionarises,namely:
Is it possible to describe the irreducible representations of G by means of the
structureofthegroupextension?
This problem was faced by many mathematicians, including I. Schur, A.H.
Clifford,and G. Mackey(who actually workedon locally compactgroups).More
recently,contributionsand expositionswere givenbyM. Isaacs, B. Huppert,Y.G.
BerkovichandE.M.Zhmud,andbyJ.M.G.FellandR.S.Doran.
Theaimofthepresentmonographistogiveanexpositionofsomeofthemany
facetsofthisproblem,atanelementarylevelandpointingtoconcreteapplications.
Indeed,therepresentationtheoryofseveralclassesofgroupsisdeterminedbythe
study of such group extensions: this is the case, for instance, for wreath products
andsemidirectproductswithAbeliannormalsubgroups;wehavedevotedanentire
monograph [28] (and several papers) to the first case and Chapter 12 of [30] to
remarkable applications of the second case (which is also widely used in the Lie
groupssetting).Actually,bothcasesareparticularexamplesofageneralprinciple,
which we call the little group method (after Mackey and E. Wigner). We give a
preciseformulationofthisprincipleandasktwofundamentalquestions:
Whenisitpossibletoapplythismethod?
When it is notpossible, is there analternative way to constructthe irreducible
representationsofthegroupextension?
The present book aims at developing powerful methods and tools, ranging
from low-dimensional group cohomology to Clifford theory and to Mackey-
Wigner theory, providing, via a thorough analysis of all such related intertwined
vii
viii Preface
concepts, a complete answer to the above questions. Our general approach—with
an emphasis on the fundamentalcontributions of Mackey, from Mackey’s lemma
(on the decompositionof ResGIndG, where H,K ≤ G) to Mackey’s obstruction
K H
(whose terminology we borrowed from Fell and Doran)—has a functional flavor
andthereforeitismoresuitableforapplicationsinaharmonicanalyticalframework.
Weclearlyhaveinmindthatanalternativeapproach,basedoncharactertheoryas
Isaacs’ [67], on the one hand gives rise, sometimes, to shorter proofsand elegant
applications to group theory, but, on the other hand, seems to be not sufficient in
ordertodevelopallthepowerfultoolsforharmonicanalysisandapplications.
Thismonographisdefinitivelynotsuitableforageneralcourseonrepresentation
theory.Indeed,thetopicstreatedaretooadvancedandpossiblytoomonothematic
forsuchageneraltarget.However,webelievethatitmightbesuccessfullyusedas
a valid supportfor such a generalcourse,for,say, a subsequentcourseon “topics
in representationtheory” as well as for some parallel “readingcourses” and/oran
“advancedseminar.”Lastbutnotleast,thematureresearchermayfinditusefulasa
referencetoCliffordtheory,Mackey’smachinery(originallydevelopedforlocally
compactgroups),andKirillov’sorbitmethod.
Itisstructuredasfollows.
InChap.1,wegivesomepreliminaryresultsonrepresentationtheoryandgroup
extensions:weprovejusttheresultswhicharenotstandard.
Chapter 2 is an exposition of Clifford theory which relates the representation
theory of a (finite) group G to that of its normal subgroups N ≤ G through the
operationsofrestriction,induction,andextension.Westudytheconjugationaction
of G (and of H = G/N) on N(cid:2), the (unitary) dual of N. Given σ ∈ N(cid:2), we
denote by I (σ) the stabilizer of σ (usually called the inertia group). We study
G
several particular cases and applications of Clifford theory, in particular we give
a precise formulation of the little group method: it may be applied when every
σ ∈ N(cid:2) has an extension (cid:3)σ to I (σ) (i.e., (cid:3)σ is an irreducible representation of
G
I (σ) and its restriction to N is σ). In this case, every irreducible representation
G
(cid:6)
of G is of the form IndG ((cid:3)σ ⊗ ψ), where ψ is an irreducible representation
IG(σ)
(cid:6)
of H (σ) := I (σ)/N and ψ ∈ I(cid:2)(σ) denotes its inflation to I (σ), that is,
G G G G
(cid:6)
ψ(h):=ψ(hN)forallh∈I (σ).
G
Chapters 3, 4, and 5 are devoted to the study of the case when A := G/N is
Abelian.Again,thereareseveralremarkablecaseswherethisconditionissatisfied,
for instance the case when N is an index two subgroup of G (which includes
G = S ,thesymmetricgroup,andN = A ,thealternatinggroup).Inthegeneral
n n
(cid:2) (cid:2)
case (Chap.3), we examine the dual action of A on G given by tensoring two
representations and its relationships with the conjugating action. In Chap.4, we
examinetheparticularcasewhenAisAbelianandeveryσ ∈ N(cid:2)hasanextension
toI (σ),sothatthelittlegroupmethodmaybeapplied.Inparticular,weexamine
G
thecasewhenAisofprimeorder.InChap.5,wediscusstherepresentationtheory
ofthealternatinggroup,ofmetacyclic,quaternionic,anddihedralgroups.Wealso
examinethecasewhenAisoforderp2.
Preface ix
Chapter6isdevotedtotheorbitmethodforstep-2nilpotentgroupsofoddorder.
When this condition is satisfied, we have a natural and powerful correspondence
between Lie rings and nilpotent groups, and this correspondence can be used to
describetheirreduciblerepresentationsofthenilpotentgroups.Themaintoolisan
actionofthegroupontheringandthecoadjointaction,namelythecorresponding
action on the dual of the Lie ring (regarded as an Abelian group). We describe
the behavior of this construction with respect to induction and restriction and we
apply it to the finite Heisenberg group. In this chapter, we also introduce some
cohomologicalmachinerythatisalsousedinthesubsequentchapters.
In Chap.7, we answer the questionsraised at the beginningof this preface:we
show that the little group method may be applied if and only if a suitable unitary
2-cocycle(theMackeyobstruction)istrivial.Wethenintroduce(unitary)projective
representations and use them to give a complete answer to the problem of the
constructionoftheirreduciblerepresentationsofafinitegroupextension.We also
examinethecommutantofaninducedrepresentationandtherelativeHeckealgebra.
Chapters 8 and 9 are devoted to the study of induction for (unitary) projective
representations:thisincludesbothMackeyandCliffordtheories.Withrespecttoour
main source(Mackey’spaperon unitaryrepresentationsof locally compactgroup
extension)wegivemoredirectproofs,involvingdetailedcocyclecomputations.
InChap.10,weexposethetheoryofprojectiverepresentationsforfiniteAbelian
groups.Someresultsaregivenundertheassumptionthatthegroupisofoddorder.
We then apply this theory in order to describe the irreducible representations of
finitemetabelian(i.e.,step-2solvable)aswellasstep-2nilpotentgroups.
Finally, in the Appendix, we review some historical aspects and discuss some
additionaltopics.
We express our deep gratitude to Pierre de la Harpe, Slava Grigorchuk, Marty
Isaacs,DanielRockmore,andJean-PierreSerreforusefulremarksandsuggestions,
as well as for their interest and encouragement. Finally, we warmly thank Elena
Griniari from Springer Verlag for her continuous care and most precious help at
variousstagesofthepreparationofthemanuscript.
Rome,Italy TullioCeccherini-Silberstein
April2022 FabioScarabotti
FilippoTolli
Contents
1 Preliminaries................................................................ 1
1.1 RepresentationsofFiniteGroups ................................... 1
1.2 TheGroupAlgebraandtheLeft-RegularRepresentation ......... 4
1.3 InducedRepresentations ............................................ 7
1.4 FurtherResultsonInducedRepresentations ....................... 11
1.5 SemidirectProducts, Wreath Products, and Group
Extensions ........................................................... 15
1.6 RegularWreathProductsandtheKaloujnine-Krasner
Theorem .............................................................. 20
2 CliffordTheory ............................................................. 27
2.1 PreliminariesandNotation .......................................... 27
2.2 BasicCliffordTheory ............................................... 30
2.3 FirstApplicationsandtheLittleGroupMethod ................... 38
2.4 TheCaseWhereA (σ):=I (σ)/N isAbelian .................. 47
G G
2.5 SomeApplicationsofMackeyTheorytoCliffordTheory ........ 56
2.6 TheG-ActionontheN-ConjugacyClasses ........................ 60
2.7 Real,Complex,andQuaternionicRepresentationsand
CliffordTheory ...................................................... 62
2.8 SemidirectProductswithanAbelianNormalSubgroup .......... 68
2.9 SemidirectProductsofAbelianGroups ............................ 69
2.10 RepresentationTheoryofWreathProductsofFinite
Groups................................................................ 74
2.11 Multiplicity-FreeNormalSubgroups ............................... 80
3 AbelianExtensions ......................................................... 81
3.1 TheDualAction ..................................................... 81
3.2 TheConjugationAction ............................................. 88
3.3 TheIntermediaryRepresentations .................................. 91
3.4 DiagrammaticSummaries ........................................... 95
xi