Representation Theory and Harmonic Analysis of Wreath Products of Finite Groups

178 Pages·2014·0.984 MB·English

by  Ceccherini-Silberstein T., Scarabotti F., Tolli F.

LONDONMATHEMATICALSOCIETYLECTURENOTESERIES
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ProfessorM.Reid,MathematicsInstitute,UniversityofWarwick,CoventryCV47AL,UnitedKingdom
Thetitlesbelowareavailablefrombooksellers,orfromCambridgeUniversityPressat
http://www.cambridge.org/mathematics
287 TopicsonRiemannsurfacesandFuchsiangroups,E.BUJALANCE,A.F.COSTA&E.MARTÍNEZ(eds)
288 Surveysincombinatorics,2001,J.W.P.HIRSCHFELD(ed)
289 AspectsofSobolev-typeinequalities,L.SALOFF-COSTE
290 QuantumgroupsandLietheory,A.PRESSLEY(ed)
291 Titsbuildingsandthemodeltheoryofgroups,K.TENT(ed)
292 Aquantumgroupsprimer,S.MAJID
293 SecondorderpartialdifferentialequationsinHilbertspaces,G.DAPRATO&J.ZABCZYK
294 Introductiontooperatorspacetheory,G.PISIER
295 Geometryandintegrability,L.MASON&Y.NUTKU(eds)
296 Lecturesoninvarianttheory,I.DOLGACHEV
297 Thehomotopycategoryofsimplyconnected4-manifolds,H.-J.BAUES
298 Higheroperands,highercategories,T.LEINSTER(ed)
299 Kleiniangroupsandhyperbolic3-manifolds,Y.KOMORI,V.MARKOVIC&C.SERIES(eds)
300 IntroductiontoMöbiusdifferentialgeometry,U.HERTRICH-JEROMIN
301 StablemodulesandtheD(2)-problem,F.E.A.JOHNSON
302 DiscreteandcontinuousnonlinearSchrödingersystems,M.J.ABLOWITZ,B.PRINARI&A.D.TRUBATCH
303 Numbertheoryandalgebraicgeometry,M.REID&A.SKOROBOGATOV(eds)
304 GroupsStAndrews2001inOxfordI,C.M.CAMPBELL,E.F.ROBERTSON&G.C.SMITH(eds)
305 GroupsStAndrews2001inOxfordII,C.M.CAMPBELL,E.F.ROBERTSON&G.C.SMITH(eds)
306 Geometricmechanicsandsymmetry,J.MONTALDI&T.RATIU(eds)
307 Surveysincombinatorics2003,C.D.WENSLEY(ed)
308 Topology,geometryandquantumfieldtheory,U.L.TILLMANN(ed)
309 Coringsandcomodules,T.BRZEZINSKI&R.WISBAUER
310 Topicsindynamicsandergodictheory,S.BEZUGLYI&S.KOLYADA(eds)
311 Groups:topological,combinatorialandarithmeticaspects,T.W.MÜLLER(ed)
312 Foundationsofcomputationalmathematics,Minneapolis2002,F.CUCKERetal(eds)
313 Transcendentalaspectsofalgebraiccycles,S.MÜLLER-STACH&C.PETERS(eds)
314 Spectralgeneralizationsoflinegraphs,D.CVETKOVIC,P.ROWLINSON&S.SIMIC
315 Structuredringspectra,A.BAKER&B.RICHTER(eds)
316 Linearlogicincomputerscience,T.EHRHARD,P.RUET,J.-Y.GIRARD&P.SCOTT(eds)
317 Advancesinellipticcurvecryptography,I.F.BLAKE,G.SEROUSSI&N.P.SMART(eds)
318 Perturbationoftheboundaryinboundary-valueproblemsofpartialdifferentialequations,D.HENRY
319 DoubleaffineHeckealgebras,I.CHEREDNIK
320 L-functionsandGaloisrepresentations,D.BURNS,K.BUZZARD&J.NEKOVÁR(eds)
321 Surveysinmodernmathematics,V.PRASOLOV&Y.ILYASHENKO(eds)
322 Recentperspectivesinrandommatrixtheoryandnumbertheory,F.MEZZADRI&N.C.SNAITH(eds)
323 Poissongeometry,deformationquantisationandgrouprepresentations,S.GUTTetal(eds)
324 Singularitiesandcomputeralgebra,C.LOSSEN&G.PFISTER(eds)
325 LecturesontheRicciflow,P.TOPPING
326 ModularrepresentationsoffinitegroupsofLietype,J.E.HUMPHREYS
327 Surveysincombinatorics2005,B.S.WEBB(ed)
328 Fundamentalsofhyperbolicmanifolds,R.CANARY,D.EPSTEIN&A.MARDEN(eds)
329 SpacesofKleiniangroups,Y.MINSKY,M.SAKUMA&C.SERIES(eds)
330 Noncommutativelocalizationinalgebraandtopology,A.RANICKI(ed)
331 Foundationsofcomputationalmathematics,Santander2005,L.MPARDO,A.PINKUS,E.SÜLI&M.J.TODD(eds)
332 Handbookoftiltingtheory,L.ANGELERIHÜGEL,D.HAPPEL&H.KRAUSE(eds)
333 Syntheticdifferentialgeometry(2ndedition),A.KOCK
334 TheNavier–Stokesequations,N.RILEY&P.DRAZIN
335 Lecturesonthecombinatoricsoffreeprobability,A.NICA&R.SPEICHER
336 Integralclosureofideals,rings,andmodules,I.SWANSON&C.HUNEKE
337 MethodsinBanachspacetheory,J.M.F.CASTILLO&W.B.JOHNSON(eds)
338 Surveysingeometryandnumbertheory,N.YOUNG(ed)
339 GroupsStAndrews2005I,C.M.CAMPBELL,M.R.QUICK,E.F.ROBERTSON&G.C.SMITH(eds)
340 GroupsStAndrews2005II,C.M.CAMPBELL,M.R.QUICK,E.F.ROBERTSON&G.C.SMITH(eds)
341 Ranksofellipticcurvesandrandommatrixtheory,J.B.CONREY,D.W.FARMER,F.MEZZADRI&N.C.SNAITH(eds)
342 Ellipticcohomology,H.R.MILLER&D.C.RAVENEL(eds)
343 AlgebraiccyclesandmotivesI,J.NAGEL&C.PETERS(eds)
344 AlgebraiccyclesandmotivesII,J.NAGEL&C.PETERS(eds)
345 Algebraicandanalyticgeometry,A.NEEMAN
346 Surveysincombinatorics2007,A.HILTON&J.TALBOT(eds)
347 Surveysincontemporarymathematics,N.YOUNG&Y.CHOI(eds)
348 Transcendentaldynamicsandcomplexanalysis,P.J.RIPPON&G.M.STALLARD(eds)
349 ModeltheorywithapplicationstoalgebraandanalysisI,Z.CHATZIDAKIS,D.MACPHERSON,A.PILLAY&A.WILKIE(eds)
350 ModeltheorywithapplicationstoalgebraandanalysisII,Z.CHATZIDAKIS,D.MACPHERSON,A.PILLAY&A.WILKIE(eds)
351 FinitevonNeumannalgebrasandmasas,A.M.SINCLAIR&R.R.SMITH
352 Numbertheoryandpolynomials,J.MCKEE&C.SMYTH(eds)
353 Trendsinstochasticanalysis,J.BLATH,P.MÖRTERS&M.SCHEUTZOW(eds)
354 Groupsandanalysis,K.TENT(ed)
355 Non-equilibriumstatisticalmechanicsandturbulence,J.CARDY,G.FALKOVICH&K.GAWEDZKI
356 EllipticcurvesandbigGaloisrepresentations,D.DELBOURGO
357 Algebraictheoryofdifferentialequations,M.A.H.MACCALLUM&A.V.MIKHAILOV(eds)
358 Geometricandcohomologicalmethodsingrouptheory,M.R.BRIDSON,P.H.KROPHOLLER&I.J.LEARY(eds)
359 Modulispacesandvectorbundles,L.BRAMBILA-PAZ,S.B.BRADLOW,O.GARCÍA-PRADA&S.RAMANAN(eds)
360 Zariskigeometries,B.ZILBER
361 Words:Notesonverbalwidthingroups,D.SEGAL
362 Differentialtensoralgebrasandtheirmodulecategories,R.BAUTISTA,L.SALMERÓN&R.ZUAZUA
363 Foundationsofcomputationalmathematics,HongKong2008,F.CUCKER,A.PINKUS&M.J.TODD(eds)
364 Partialdifferentialequationsandfluidmechanics,J.C.ROBINSON&J.L.RODRIGO(eds)
365 Surveysincombinatorics2009,S.HUCZYNSKA,J.D.MITCHELL&C.M.RONEY-DOUGAL(eds)
366 Highlyoscillatoryproblems,B.ENGQUIST,A.FOKAS,E.HAIRER&A.ISERLES(eds)
367 Randommatrices:Highdimensionalphenomena,G.BLOWER
368 GeometryofRiemannsurfaces,F.P.GARDINER,G.GONZÁLEZ-DIEZ&C.KOUROUNIOTIS(eds)
369 Epidemicsandrumoursincomplexnetworks,M.DRAIEF&L.MASSOULIÉ
370 Theoryofp-adicdistributions,S.ALBEVERIO,A.YU.KHRENNIKOV&V.M.SHELKOVICH
371 Conformalfractals,F.PRZYTYCKI&M.URBANSKI
372 Moonshine:Thefirstquartercenturyandbeyond,J.LEPOWSKY,J.MCKAY&M.P.TUITE(eds)
373 Smoothness,regularityandcompleteintersection,J.MAJADAS&A.G.RODICIO
374 Geometricanalysisofhyperbolicdifferentialequations:Anintroduction,S.ALINHAC
375 Triangulatedcategories,T.HOLM,P.JØRGENSEN&R.ROUQUIER(eds)
376 Permutationpatterns,S.LINTON,N.RUŠKUC&V.VATTER(eds)
377 AnintroductiontoGaloiscohomologyanditsapplications,G.BERHUY
378 Probabilityandmathematicalgenetics,N.H.BINGHAM&C.M.GOLDIE(eds)
379 Finiteandalgorithmicmodeltheory,J.ESPARZA,C.MICHAUX&C.STEINHORN(eds)
380 Realandcomplexsingularities,M.MANOEL,M.C.ROMEROFUSTER&C.T.CWALL(eds)
381 Symmetriesandintegrabilityofdifferenceequations,D.LEVI,P.OLVER,Z.THOMOVA&P.WINTERNITZ(eds)
382 Forcingwithrandomvariablesandproofcomplexity,J.KRAJÍCEK
383 Motivicintegrationanditsinteractionswithmodeltheoryandnon-ArchimedeangeometryI,R.CLUCKERS,J.NICAISE&
J.SEBAG(eds)
384 Motivicintegrationanditsinteractionswithmodeltheoryandnon-ArchimedeangeometryII,R.CLUCKERS,J.NICAISE&
J.SEBAG(eds)
385 EntropyofhiddenMarkovprocessesandconnectionstodynamicalsystems,B.MARCUS,K.PETERSEN&T.WEISSMAN(eds)
386 Independence-friendlylogic,A.L.MANN,G.SANDU&M.SEVENSTER
387 GroupsStAndrews2009inBathI,C.M.CAMPBELLetal(eds)
388 GroupsStAndrews2009inBathII,C.M.CAMPBELLetal(eds)
389 Randomfieldsonthesphere,D.MARINUCCI&G.PECCATI
390 Localizationinperiodicpotentials,D.E.PELINOVSKY
391 Fusionsystemsinalgebraandtopology,M.ASCHBACHER,R.KESSAR&B.OLIVER
392 Surveysincombinatorics2011,R.CHAPMAN(ed)
393 Non-abelianfundamentalgroupsandIwasawatheory,J.COATESetal(eds)
394 Variationalroblemsindifferentialgeometry,R.BIELAWSKI,K.HOUSTON&M.SPEIGHT(eds)
395 Howgroupsgrow,A.MANN
396 Arithmeticdfferentialoperatorsoverthep-adicintegers,C.C.RALPH&S.R.SIMANCA
397 Hyperbolicgeometryandapplicationsinquantumchaosandcosmology,J.BOLTE&F.STEINER(eds)
398 Mathematicalmodelsincontactmechanics,M.SOFONEA&A.MATEI
399 Circuitdoublecoverofgraphs,C.-Q.ZHANG
400 Densespherepackings:ablueprintforformalproofs,T.HALES
401 AdoubleHallalgebraapproachtoaffinequantumSchur–Weyltheory,B.DENG,J.DU&Q.FU
402 Mathematicalaspectsoffluidmechanics,J.ROBINSON,J.L.RODRIGO&W.SADOWSKI(eds)
403 Foundationsofcomputationalmathematics:Budapest2011,F.CUCKER,T.KRICK,A.SZANTO&A.PINKUS(eds)
404 Operatormethodsforboundaryvalueproblems,S.HASSI,H.S.V.DESNOO&F.H.SZAFRANIEC(eds)
405 Torsors,étalehomotopyandapplicationstorationalpoints,A.N.SKOROBOGATOV(ed)
406 Appalachiansettheory,J.CUMMINGS&E.SCHIMMERLING(eds)
407 Themaximalsubgroupsofthelow-dimensionalfiniteclassicalgroups,J.BRAY,D.HOLT&C.RONEY-DOUGAL
LondonMathematicalSocietyLectureNoteSeries:410
Representation Theory and Harmonic
Analysis of Wreath Products
of Finite Groups
TULLIO CECCHERINI-SILBERSTEIN
UniversitàdelSannio,Italy
FABIO SCARABOTTI
SapienzaUniversitàdiRoma,Italy
FILIPPO TOLLI
UniversitàRomaTre,Italy
UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom
PublishedintheUnitedStatesofAmericabyCambridgeUniversityPress,NewYork
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©T.Ceccherini-Silberstein,F.ScarabottiandF.Tolli2014
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Ceccherini-Silberstein,Tullio.
Representationtheoryandharmonicanalysisofwreathproductsoffinitegroups/
TullioCeccherini-Silberstein,FabioScarabotti,andFilippoTolli.
pages cm
Includesbibliographicalreferencesandindex.
ISBN978-1-107-62785-7(Paperback)
1. Harmonicanalysis. 2. Finitegroups.
I. Scarabotti,Fabio. II. Tolli,Filippo,1968– III. Title.
QA403.C442014
512(cid:2).23–dc23 2013024946
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Tomyparents,Cristina,Nadiya,andVirginia
Tothememoryofmyfather,andtomymother

Contents
Preface pagexi
1 Generaltheory 1
1.1 Inducedrepresentations 1
1.1.1 Definitions 1
1.1.2 Transitivityandadditivityofinduction 8
1.1.3 Frobeniuscharacterformula 10
1.1.4 Inductionandrestriction 11
1.1.5 Inducedrepresentationsandinducedoperators 14
1.1.6 Frobeniusreciprocity 14
1.2 Harmonicanalysisonafinitehomogeneousspace 16
1.2.1 Frobeniusreciprocityforpermutationrepresentations 16
1.2.2 Sphericalfunctions 22
1.2.3 TheothersideofFrobeniusreciprocityfor
permutationrepresentations 34
1.2.4 Gelfandpairs 37
1.3 Cliffordtheory 41
1.3.1 Cliffordcorrespondence 42
1.3.2 Thelittlegroupmethod 49
1.3.3 Semidirectproducts 50
1.3.4 SemidirectproductswithanAbeliannormalsubgroup 51
1.3.5 Theaffinegroupoverafinitefield 52
1.3.6 ThefiniteHeisenberggroup 56
2 Wreathproductsoffinitegroupsandtheir
representationtheory 60
2.1 Basicpropertiesofwreathproductsoffinitegroups 60
2.1.1 Definitions 60
vii
viii Contents
2.1.2 Compositionandexponentiationactions 63
2.1.3 Iteratedwreathproductsandtheiractionson
rootedtrees 67
2.1.4 Spherically homogeneous rooted trees and their
automorphismgroup 69
2.1.5 Thefiniteultrametricspace 70
2.2 Twoapplicationsofwreathproductstogrouptheory 76
2.2.1 ThetheoremofKaloujnineandKrasner 76
2.2.2 Primitivityoftheexponentiationaction 78
2.3 Conjugacyclassesofwreathproducts 81
2.3.1 Ageneraldescriptionofconjugacyclasses 82
2.3.2 ConjugacyclassesofgroupsoftheformC (cid:3)G 86
2
2.3.3 ConjugacyclassesofgroupsoftheformF (cid:3)S 89
n
2.4 Representationtheoryofwreathproducts 92
2.4.1 Theirreduciblerepresentationsofwreathproducts 92
2.4.2 The character and matrix coefficients of the
representation(cid:2)σ 95
2.5 RepresentationtheoryofgroupsoftheformC (cid:3)G 98
2
2.5.1 Representation theory of the finite lamplighter
groupC (cid:3)C 99
2 n
2.5.2 Representationtheoryofthehyperoctahedralgroup
C (cid:3)S 100
2 n
2.6 RepresentationtheoryofgroupsoftheformF (cid:3)S 101
n
2.6.1 RepresentationtheoryofS (cid:3)S 103
m n
3 Harmonicanalysisonsomehomogeneousspacesoffinite
wreathproducts 104
3.1 Harmonicanalysisonthecompositionoftwo
permutationrepresentations 104
3.1.1 Decompositionintoirreduciblerepresentations 104
3.1.2 Sphericalmatrixcoefficients 107
3.2 ThegeneralizedJohnsonscheme 110
3.2.1 TheJohnsonscheme 110
3.2.2 Thehomogeneousspace(cid:3) 112
h
3.2.3 Twospecialkindsoftensorproduct 117
3.2.4 The decomposition of L((cid:3) ) into irreducible
h
representations 120
3.2.5 Thesphericalfunctions 123
3.2.6 ThehomogeneousspaceV(r,s)andtheassociated
Gelfandpair 127