Q-systems as cluster algebras II Cartan matrix of

arXiv:0803.0362v2[math.RT]17Jul2008Q-SYSTEMSASCLUSTERALGEBRASII:CARTANMATRIXOFFINITETYPEANDTHEPOLYNOMIALPROPERTYP.DIFRANCESCOANDR.KEDEMAbstract.WedefinetheclusteralgebraassociatedwiththeQ-systemfortheKirillov-ReshetikhincharactersofthequantumaffinealgebraUq(bg)foranysimpleLiealgebrag,generalizingthesimply-lacedcasetreatedin[Kedem2007].Wedescribesomespecialpropertiesofthisclusteralgebra,andexplainitsrelationtothedeformedQ-systemswhichappearedonourproofofthecombinatorial-KRconjecture.Weprovethatthepolynomialityoftheclustervariablesintermsofthe“initialclusterseeds”,includingsolutionsoftheQ-system,isaconsequenceoftheLaurentphenomenonandtheboundaryconditions.WealsogiveaformulationofbothQ-systemsandgen-eralizedT-systemsasclusteralgebraswithcoefficients.ThisprovidesaproofofthepolynomialityofsolutionsofgeneralizedT-systemswithappropriateboundaryconditions.1.IntroductionTheQ-systemisarecursionrelationforthecharactersofcertainfinite-dimensionalrepresen-tationsofthequantumaffinealgebraUq(bg)ortheYangianY(g),wheregisasimpleLiealgebra.Q-systemswereintroducedin[15]fortheclassicalalgebras.Theywerelatergeneralizedby[10]fortheexceptionalalgebrasandlatertomorecomplicatedsituations,suchastwistedquantumaffinealgebras[11]anddoubleaffinealgebras[12].ThespecialmodulesrelatedtoQ-systemsarecalled[17,16]Kirillov-Reshetikhinmodules.ThefactthattheircharacterssatisfytheQ-systemwasprovedbyKirillovandReshetikhin[15]forg=Ar,byNakajima[18]forthesimply-lacedalgebrasandbyHernandez[13,12]infurthergenerality.ClusteralgebraswereintroducedbyFominandZelevinsky[4]in2000,andareaverygeneralalgebraictoolwhichhassincebeenappliedinvariousalgebraic,combinatorialandgeometriccontexts.Inparticular,theyhavebeenusedtostudyY-systems[6],whicharerelatedtoQ-systemsinthesensethatbothcanbederivedfromT-systems[17].TheT-systemsareaconsequenceofthefusionrelationsforYangianorquantumalgebramodules.TheformoftheQ-systemsuggeststhatitshouldbepossibletorecastitaspartofaclusteralgebra.Thefirststepinthisreformulation,forthecasewheregissimply-laced,wasderivedin[14].Inthecurrentarticle,wegiveageneralizationofthiscasetonon-simplylacedg.Inthisformulation,theQ-systemappearsinaverysimpleandeasilygeneralizableform.WenotethatitdoesnotappeartobedirectlyrelatedtotheclusteralgebracomingfromtheY-systemstudiedbyFominandZelevinsky[6].In[3],weprovedacombinatorialidentity(“theM=N”conjectureof[10])whichimpliestheproofofthecombinatorialKirillov-Reshetikhinconjecture.Inourproof,weintroducedwhatwecalledthedeformedQ-system,dependingonanincreasingnumberofformalvariables.AspecializationofthissystemcanbeexpressedastheQ-systemwithgeneralboundaryconditionsDate:July17,2008.12P.DIFRANCESCOANDR.KEDEM(notcorrespondingtoKirillov-Reshetikhincharacters),orasaclusteralgebra.Itisessentialfortheformulationasaclusteralgebratohavethismoregeneralsystem.OurproofoftheM=NidentitydependscruciallyonthefactthattheKR-charactersarepolynomialsinthefundamentalKR-characters.Thiscanberephrasedintermsoftheclusteralgebra,byconsideringaspecializationoftheinitialclustervariablestothespecialpointwhichgivestheKR-charactersattheothernodesoftheclustergraph.WecallthisspecializationoftheinitialparameterstheKRpoint.Ontheotherhand,itisknownthatclustervariablesobeytheLaurentphenomenon[5].Weshowthat,underthespecializationoftheclustervariablestotheKRpoint,thisbecomeswhatwecalledin[14]thestrongLaurentphenomenon.Thatis,theclustervariablesontheentireclustergraph,notjustthesubgraphcorrespondingtotheQ-system,arepolynomialsintheinitialclustervariables.Thepaperisorganizedasfollows.WerecallthedefinitionofQ-systemsforanysimpleLiealgebrainSection2,aswellasthedefinitionofnormalizedclusteralgebraswithoutcoefficients.Section3dealswiththeformulationofeachQ-systemasasubgraphinaclusteralgebra.InSection3.1,wereviewtheresultsof[14]abouttheformulationofQ-systemsasclusteralgebrasforsimply-lacedLiealgebras.InSections3.2and3.3,weformulatetheclusteralgebrascorrespondingtothenon-simplylacedsimpleLiealgebras.InSection4,weprovethatthespecialboundaryconditionswhichgivesolutionsoftheQ-systemascharactersofKirillov-Reshetikhinmodulesimplythepolynomialityoftheclustervariablesasfunctionsoftheseedvariablesattheboundarynode.Section5isadiscussionoftheresults.AppendixAisareformulationoftheQ-systemasaclusteralgebrawithcoefficients,addressingthetechnicalpointofsubtraction-freeexpressions.Inthebodyofthepaper,thisisdonethrougharenormalizationoftheclustervariables,whichisnotalwaysgeneralizable.AppendixBisadiscussionoftheformulationofgeneralizedT-systemsasclusteralgebras,withtwomainexamples,bothofwhichhavethepolynomialityproperty.Acknowledgements.WethankBernhardKeller,HughThomasandespeciallySergeiFominfortheirvaluableinput.RKthanksCEA-SaclayIPhTfortheirhospitality.WethanktheorganizersoftheMSRIprogramon“CombinatorialRepresentationTheory”fortheirhospitality.RKissupportedbyNSFgrantDMS-05-00759.PDFacknowledgesthesupportofEuropeanMarieCurieResearchTrainingNetworksENIGMAMRT-CT-2004-5652,ENRAGEMRTN-CT-2004-005616,ESFprogramMISGAM,andofANRprogramGIMPANR-05-BLAN-0029-01.2.Definitions2.1.TheQ-system.2.1.1.KR-