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1、TopicsinRepresentationTheory:SU(2)RepresentationsandTheirApplicationsWe’vesofarbeenstudyingaspeci?crepresentationofanarbitrarycompactLiegroup,theadjointrepresentation.Therootsaretheweightsofthisrepre-sentation.Wewouldnowliketobeginthestudyofarbitaryrepresentationsandtheirwei
2、ghts.Anarbitrary?nitedimensionsionalrepresesentationwillhaveadirectsumdecompositionMV=Vααwheretheαaretheweightsoftherepresentationlabelledbyelementsoft?,andVαistheα-weightspace,i.e.thevectorsvinVsatisfyingHv=α(H)vforH∈t.ThedimensionofVαiscalledthemultiplicityofα.Theproblemwe
3、wanttosolveforeachcompactLiegroupGistoidentifytheirreduciblerepresentations,computingtheirweightsandmultiplicities.Animportantrelationbetweenrootsandweightsisthefollowing:Lemma1.IfX∈gβ,thenitmapsX:Vα→Vα+βProof:Ifv∈Vα,H∈tHXv=XHv+[H,X]v=Xα(H)v+β(H)Xv=(α(H)+β(H))Xvsotherootsact
4、onthesetofweightsbytranslation.Wewillbeginwiththesimplestcase,thatofG=SU(2).Thiscaseisofgreatimportancebothasanexampleofallthephenomenawewanttostudyforhigherrankcases,aswellasplayingafundamentalpartitselfintheanalysisofthegeneralcase.1ReviewofSU(2)RepresentationsOnereasontha
5、tSU(2)representationsareespeciallytractableisthatthereisasimpleexplicitconstructionoftheirreduciblerepresentations.ConsiderthespaceVnofhomogeneouspolynomialsoftwocomplexvariables.Anelementof2thisspaceisoftheformf(z,z)=azn+azn?1z+···+azn1201112n2ThegroupSU(2)actsonVnthroughth
6、eactionofU∈SU(2)asalinear2transformationonthevectorz=(z1,z2)asfollowsπ(U)f(z)=f(U?1z)1Thisisagrouphomomorphismsince?1?1?1π(U1)(π(U2)f)(z)=(π(U2)f)(U1z)=f(U2U1z)=π(U1U2)f(z)TherepresentationonVnisofdimensionn+1andonecanshowthatitis2irreducible.Bydi?erentiatingtheactionofthegr
7、ouponecanexplicitlygettheactionoftheLiealgebraandone?ndsthat?f?fπ?(H)f=?z1+z2?z1?z2+?fπ?(X)f=?z2?z1??fπ?(X)f=?z1?z2OnecanexplicitlyworkouthowtheLiealgebraactsonVn.Notethat2actingonthemonomialswe?ndπ(H)zjzk=(?j+k)zjzk?1212π(X+)zjzk=?jzj?1zk+1?1212π(X?)zjzk=?kzj+1zk?1?1212Them
8、onomialsareeigenvectorsofπ?(H)witheigenvalues?n,?n+2,···,n?2,n,thesearethew
