discrete groups lecture notes

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1、DISCRETEGROUPS1version7/4/08InstituteforTheoreticalPhysicsUtrechtUniversityBetaFaculty2008BasedonlecturenotesbyR.Loll(inturnbasedinpartonlecturenotesbyD.Vvedensky(ImperialCollegeLondon))withmodi¯cationsandadditionsbyG.'tHooftandH.SahlmannContents1Introduction12BasicNotion

2、sofGroupTheory42.1Thesymmetricgroup..............................72.2Cosets......................................92.3NormalSubgroups,QuotientGroups.....................112.4Conjugacyclasses................................153Representationsof¯nitegroups173.1(Ir)reduciblere

3、presentations..........................193.2Unitaryrepresentations.............................213.3Schur'sLemma(s)................................253.4TheGreatOrthogonalityTheorem......................273.5Characters....................................313.6Theregularrepr

4、esentation...........................363.7Charactertables.................................394Somemoreapplications404.1Crystallography.................................404.2Bloch'sTheorem................................421.IntroductionSymmetriesareoperationsunderwhichthesys

5、temunderinvestigationremainsunchanged.Symmetriesareextremelyimportantinphysics:Ontheonehandthisisbecausenatureseemstoobeycertainsymmetries.Ontheotherhandtherearemanysituationsinwhichasymmetryispresentatleasttoagoodapproximation.Inanyofthesecases,thesym-metriesprovideapowe

6、rfultoolfortheanalysisofthesystem.Wewillgivesomeexamplesbelow.Inmanycases,symmetryoperationsactuallyformgroups:GiventwosymmetriesS1andS2,onecanapplythemoneaftertheother,andobtainanewsymmetrytransfor-mationS3=S2S1.And,sincesymmetriesshouldnotdestroyinformation"toagivensym

7、metryoperationS,thereisone,calledS¡1thatundoes"thetransformationS.Aswewillsee,thesearetheessentialingredientsinthede¯nitionofagroup.Letusgivesomeexamplesofsymmetries.1)Themostintuitivesymmetriesarethoseofgeometricobjects.Thesearetransfor-mations(likerotationsorre°ections

8、)thatleaveageometricobjectinvariant.Thesearesometimescalledisometries.Asanexample,inFigure1vario