Topics in Representation Theory SU

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1、TopicsinRepresentationTheory:SU(2)RepresentationsandTheirApplicationsWe’vesofarbeenstudyingaspecificrepresentationofanarbitrarycompactLiegroup,theadjointrepresentation.Therootsaretheweightsofthisrepre-sentation.Wewouldnowliketobeginthestudyofarbitaryrepresentationsandtheirwei

2、ghts.AnarbitraryfinitedimensionsionalrepresesentationwillhaveadirectsumdecompositionMV=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.Bydifferentiatingtheactionofthegr

7、ouponecanexplicitlygettheactionoftheLiealgebraandonefindsthat∂f∂fπ∗(H)f=−z1+z2∂z1∂z2+∂fπ∗(X)f=−z2∂z1−∂fπ∗(X)f=−z1∂z2OnecanexplicitlyworkouthowtheLiealgebraactsonVn.Notethat2actingonthemonomialswefindπ(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