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1、InternationalJournalofTheoreticalPhysics,Vol.15,No.3(1976),pp.201-206OntheWeylCharacterFormulaforSU(n)R.J.PLYMENDepartmentofMathematics,UniversityofManchester,ManchesterM139PL,EnglandReceived:19February1975AbstractInthisNote,wedrawastraightlinebetweenthere
2、presentationtheoryofSU(3)andtheSU(3)-classificationschemesinparticlephysics.OurapproachisbasedonthatofWeyl,butwehaveinmindtheversionswhichappear,"inmoderndress,"inAdamsandBott.Ourformulationbringsanimportantpartofparticlephysicsintolinewithtwocon-temporary
3、accountsofcompactLiegroups.1.TheGroupSU(3)WebeginwithacelebratedformuladuetoWeyl(1950,p.381).LetSU(3)bethegroupofall3x3unitarymatriceswithdeterminant1.LetM(e)=diag(el,e2,ca)beadiagonalmatrixinSU(3).Thuslqi=le21=lea[=1=ele2e3.LetUbeanirreduciblerepresentati
4、onofSU(3)onafinite-dimensionalcomplexvectorspace,andletXbeitscharacter.ThusX(M)=Tr(U(M)),MESU(3).Let~1rG1glter,es,ll=e2re2stefed1wherer,sarepositiveintegers.TheWeylformulafortheirreduciblecharactersXr,sofSU(3)is×,,s(M(e))=ler,d,1I/leZ,e,11r>sItfollowsthatt
5、heirreduciblecharactersaresymmetricpolynomialsinq,e2,ca.Hereisatableoffiveusefulcharacters:(1)Xx,I(M(e))=1(2)Xa,I(M(e))=el+e2+e3©1976PlenumPublishingCorporation.Nopartofthispublicationmaybereproduced,storedinaretrievalsystem,ortransmitted,inanyformorbyanym
6、eans,electronic,mechanical,photocopying,microfilming,recording,orotherwise,withoutwrittenpermissionofthepublisher.201202R.J.PLYMEN(3)Xa,2(M(e))=e2e3+e3el+ele2(4)X4,2(M(e))=2+eae22+eael2+ele32+ele22+e2el2+e2e32(5)×s,l(M(e))=l+ela+e2a+%a+eae2Z+eae~2+eaea2+el
7、e22+e2e122-[-e2e3Noweverysymmetricpolynomial(withintegercoefficients)inel,e2,e3isapolynomialintheelementarysymmetricfunctionse~+ez+%,e:%+eae~+ele2,ele2%=1.Thisisthemathematicalbasisoftheclaimthat"thequarksandantiquarksgenerateallSU(3)multiplets."2.TheSubgr
8、oupTThesubgroupTofdiagonalmatricesisclearlycommutative,henceitsirreduciblerepresentationsareone-dimensional.Werecallthataone-dimensionalrepresentationisidenticalwithitscharacter.LetXr,~determinetheirreducible
