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
2IGORB.FRENKELANDALISTAIRSAVAGE
of[DJKMO2]ontheconstructionofabasisofa neLiealgebrarepresentations.Atthesametime,wegiveasimpleparametrizationoftheirreduciblecomponentsofNakajimaquivervarietiesassociatedtoin niteandcyclicquivers.
Thecomparisonofthetwoverydi erenttheoriesbringssomesurprisesandsuggestsinterestingnewdirections.Inparticular,theYoungdiagramsthatareroutinelyusedinrepresentationtheoryoftypeALiealgebrasacquireanexplicitgeometricmeaning:Theypicturepreciselyrepresentationsofthecorrespondingquiverssatisfyingastabilityconditionforlevel1(seeFigure2inthetext).Ontheotherhand,thealgebraicconstructionsof[DJKMO2]involvesubstantiallythe n+1,whicharenotdirectlycoveredbyNaka-highestweightrepresentationsofgl
jima’stheory.Wede nenewvarietiesbyrelaxingthenilpotencyconditioninthede nitionofNakajima’squivervarietiesandshowthattheirreduciblecomponentsofthesenewvarietiesareinone-to-onecorrespondencewithbasesofthehighest n+1.Wealsomentionsomeinterestingproblemsthatweightrepresentationsofgl
ariseasaresultofthecomparisionofgeometricandalgebraicconstructions.
Westronglybelievethatthemainresultsofthecurrentpaperre ectaverygeneralprinciplethatassertstheprofoundgeometricorgaugetheoreticoriginofvariousalgebraicandcombinatorialstructuresofintegrablemodelsinstatisticalmechanics.Therelationofbothsubjectstotherepresentationtheoryofa neLiealgebrasisanecessaryprerequisiteofthisprinciple.Howeverweexpectmuchmore;namelythatvariousspeci cconstructionsappearinginintegrablemodelsofstatisticalmechanicsthatincludetensorproducts,fusionproducts,branchingrules,Bethe’sansatzandtheYang-Baxterequationitselfre ectcertaingeometricfactsaboutNakajimavarieties,Malkin-Nakajimatensorproductvarieties,variousLagrangiansubvarietiesandcorrespondinggaugetheoriesoncommutativeand,possibly,noncommutativespaces.Thepresentpaperisasmallbutindicativesteptowardthisvastprogram.
Thepaperisorganizedasfollows.InSection1werecallthede nitionofLusztig’squivervarietiesandcharacterizationsoftheirreduciblecomponentsintypesA∞(1)andAn.WealsointroduceaversionofLusztig’squivervarietiesfortheLiealge- bragln+1.Section2containsthede nitionofNakajima’squivervarietiesandtheLiealgebraactiononasuitablespaceofconstructiblefunctionsonthesevarietiesisgiveninSection3.InSection4wegiveanenumerationoftheirreduciblecompo-nentsofthequivervarietiesforlevel1intermsofYoungdiagrams.WealsoidentifythegeometricactionofthetypeA∞LiealgebrainthebasisenumeratedbyYoungdiagrams.InSection5weextendtheenumerationoftheirreduciblecomponentsofthequivervarietiestoarbitrarylevelandweestablishamatchwiththeindexingofbasesofthecorrespondingrepresentationscomingfromstatisticalmechanics.Finally,inSection6,wecomparetheweightstructureofthebasesresultingfromquivervarietiesandthepathrealizationsofstatisticalmechanicsandmakecertainoftheircompletecoincidence.
Theresearchofthe rstauthorwassupportedinpartbytheNationalScienceFoundation(NSF).TheresearchofthesecondauthorwassupportedinpartbytheNaturalSciencesandEngineeringResearchCouncilofCanada(NSERC).
1.Lusztig’sQuiverVarieties
Inthissection,wewillrecounttheexplicitdescriptiongivenin[L1]oftheirre-
(1)duciblecomponentsofLusztig’squivervarietyinthecaseoftypesA∞andAn.
Seethisreferencefordetails,includingproofs.
LetIbeasetofverticesoftheDynkingraphofaKac-MoodyLiealgebragandletHbethesetofpairsconsistingofanedgetogetherwithanorientationofit.Forh∈H,letin(h)(resp.out(h))betheincoming(resp.outgoing)vertexof