群论和物理学（英文版）

Grouptheoryisoneofthegreatachievementsof19thcenturymathematics.Itemergedasaunifyingideadrawingonfourdifferentsources：numbertheory，thetheoryofequations，geometry，andcrystallography.TheearlymotivationfromnumbertheorystemmedfromtheworkofEuler，LegendreandGaussonpowerresidues.Inthetheoryofequations，thestudyofvariouspermutationgroupsbecameincreasinglyimportantthroughtheworkofLagrange，Ruffini，Gauss，Abel，Cauchy，andespeciallyGalois.Thediscoveryofnewtypesofgeometries-includingnon-Euclidean，affine，projectiveetc.-led，eventually，tothefamousErlangenprogramofKlein，whichproposedthatthetruestudyofanygeometryliesinananalysisofitsgroupofmotions.Incrystallography，thepossiblesymmetriesoftheinternalstructureofacrystalwereenumeratedlongbeforetherewasanypossibilityofitsphysicaldetermination（byX-rayanalysis）.

Preface
1Basicdefinitionsandexamples
1.1Groups:definitionandexamples
1.2Homomorphisms:therelationbetweenSL2,andtheLorentzgroup
1.3Theactionofagrouponaset
1.4Conjugationandconjugacyclasses
1.5Applicationstocrystallography
1.6ThetopologyofSU2andSO3
1.7Morphisms
1.8TheclassificationofthefinitesubgroupsofSO3
1.9TheclassificationofthefinitesubgroupsofO3
1.10Theicosahedralgroupandthefullerenes

2Representationtheoryoffinitegroups
2.1Definitions,examples,irreducibility
2.2Completereducibility
2.3Schur'slemma
2.4Charactersandtheirorthogonalityrelations
2.5Actiononfunctionspaces
2.6Theregularrepresentation
2.7Charactertables
2.8Therepresentationsofthesymmetricgroup

3Molecularvibrationsandhomogeneousvectorbundles
3.1Smalloscillationsandgrouptheory
3.2Moleculardisplacementsandvectorbundles
3.3Inducedrepresentations
3.4Principalbundles
3.5Tensorproducts
3.6Representativeoperatorsandquantummechanicalselectionrules
3.7Thesemiclassicaltheoryofradiation
3.8Semidirectproductsandtheirrepresentations
3.9Wigner'sclassificationoftheirreduciblerepresentationsofthePoincaregroup
3.10Parity
3.11TheMackeytheoremsoninducedrepresentations,withapplicationstothesymmetricgroup
3.12Exchangeforcesandinducedrepresentations

4CompactgroupsandLiegroups
4.1Haarmeasure
4.2ThePeter-Weyltheorem
4.3TheirreduciblerepresentationsofSU2
4.4TheirreduciblerepresentationsofSO3andsphericalharmonics
4.5Thehydrogenatom
4.6Theperiodictable
4.7Theshellmodelofthenucleus
4.8TheClebsch-Gordancoefficientsandisospin
4.9Relativisticwaveequations
4.10Liealgebras
4.11Representationsofsu2

5TheirreduciblerepresentationsofSUn
5.1TherepresentationofGlVonther-foldtensorproduct
5.2GlVspansHornsrTrV,TrV
5.3DecompositionofTrVintoirreducibles
5.4Computationalrules
5.5DescriptionoftensorsbelongingtoW
5.6RepresentationsofGlVandSlVonU
5.7Weightvectors
5.8Determinationoftheirreduciblefinite-dimensionalrepre-sentationsofSld,C
5.9Strangeness
5.10Theeight-foldway
5.11Quarks
5.12Colorandbeyond
5.13Wheredowestand

AppendixATheBravaislatticesandthearithmeticalcrystalclasses
A.1Thelatticebasisandtheprimitivecell
A.2The14Bravaislattices
AppendixBTensorproduct
AppendixCIntegralgeometryandtherepresentationsofthesymmetricgroup
C.1Partitionpairs
C.2Proofofthemaincombinatoriallemma
C.3TheLittlewood-RichardsonruleandYoung'srule
C.4TheringofvirtualrepresentationsofalltheSn
C.5Dimensionformulas
C.6TheMurnaghan-Nakayamarule
C.7CharactersofGlV
AppendixDWigner'stheoremonquantummechanicalsymmetries
AppendixECompactgroups,Haarmeasure,andthePeter-Weyltheorem
AppendixFAhistoryof19thcenturyspectroscopy
AppendixGCharactersandfixedpointformulasforLiegroups
Furtherreading
Index