Bases of representations of type A affine Lie algebras via q

We relate two apparently different bases in the representations of affine Lie algebras of type A: one arising from statistical mechanics, the other from gauge theory. We show that the two are governed by the same combinatorics and therefore can be viewed a

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aBASESOFREPRESENTATIONSOFTYPEAAFFINELIEALGEBRASVIAQUIVERVARIETIESANDSTATISTICALMECHANICSIGORB.FRENKELANDALISTAIRSAVAGEAbstract.Werelatetwoapparentlydi erentbasesintherepresentationsofa neLiealgebrasoftypeA:onearisingfromstatisticalmechanics,theotherfromgaugetheory.Weshowthatthetwoaregovernedbythesamecombina-toricsthatalsorespectstheweightspacedecompositionoftherepresentations.Inparticular,weareabletogiveanalternativeandmuchsimplergeometricproofofthemainresultof[DJKMO2]ontheconstructionofbasesofa neLiealgebrarepresentations.Atthesametime,wegiveasimpleparametrizationoftheirreduciblecomponentsofNakajimaquivervarietiesassociatedtoin niteandcyclicquivers.Wealsode nenewvarietieswhoseirreduciblecomponentsareinone-to-onecorrespondencewithbasesofthehighestweightrepresenta-tionsofgl n+1.IntroductionAremarkablerelationbetweenrepresentationtheoryofa neLiealgebrasandmodelsofstatisticalmechanicsbasedontheYang-Baxterequationhasbeendis-coveredandintensivelystudiedbyE.Date,M.Jimbo,A.Kuniba,T.MiwaandM.Okado(see[DJKMO1,DJKMO2]andreferencestherein).Oneoftheimportant ndingsoftheaboveauthorsisthattheone-dimensionalcon gurationsumsforthesemodelsgiverisetocharactersofintegrablehighestweightrepresentationsofa neLiealgebras.Thisrelationyieldscertainexplicitbasesintherepresenta-tionsthatadmitpurecombinatorialdescriptionsandimplyvariousidentitiesforthecharacters.Anotherastonishingrelationbetweenrepresentationtheoryofa neLiealgebrasandmodulispacesofsolutionsofself-dualYang-Millsequationshasbeenaccom-plishedbyH.Nakajima[N1,N3],whoobservedaprofoundlinkbetweenhisearlierworkwithP.KronheimerandtheresultsofG.Lusztig[L1,L2].AttheheartofbothworksthatprecededtheNakajimadiscoveryarequivervarietiesassociatedwithextendedDynkindiagrams.Nakajimaintroducedaspecialclassofquivervarietiesassociatedwithintegrablehighestweightrepresentationsofa neLieal-gebrasandobtainedageometricdescriptionoftheaction.Healsode nedcertain

Lagrangiansubvarietieswhoseirreduciblecomponentsyieldageometricbasisofthea neLiealgebrarepresentations.

Thecentralgoalofthepresentpaperistorelatethetwoapparentlydi erentbasesintherepresentationsofa neLiealgebrasoftypeA:onearisingfromstatis-ticalmechanics,theotherfromgaugetheory.Weshowthatthetwoaregovernedbythesamecombinatoricsthatalsorespectstheweightspacedecompositionoftherepresentations.Thisidenti cationallowsonetogiveanaturalconceptualframe-worktotheintricatestructureofstatisticalmechanicalmodelsandalsotomakeexplicitcalculationsinaseeminglyintractablegeometricsetting.Inparticular,weareabletogiveanalternativeandmuchsimplergeometricproofofthemainresult