Crystal bases and simple modules for Hecke algebra

arXiv:math/0506555v3[math.RT]20Feb2007CrystalbasesandsimplemodulesforHeckealgebrasoftypeG(p,p,n)∗JunHuDepartmentofAppliedMathematicsBeijingInstituteofTechnologyBeijing,100081,P.R.ChinaE-mail:[e-mail]talbasistheoryforFockspacesoverquantumaffinealgebrastothemodularrepresentationsofthecyclotomicHeckealgebrasoftypeG(p,p,n).Thisyieldsaclassificationofsimplemod-ulesoverthesecyclotomicHeckealgebrasinthenon-separatedcase,generalizingourpreviouswork[J.Hu,J.Algebra267(2003)7-20].Theseparatedcasewascompletedin[J.Hu,J.Algebra274(2004)446–490].Furthermore,weuseNaito–Sagaki’swork[S.Naito&D.Sagaki,J.Algebra251(2002)461–474]onLakshmibai–SeshadripathsfixedbydiagramautomorphismstoderiveexplicitformulasforthenumberofsimplemodulesovertheseHeckealgebras.Theseformu-lasgeneralizeearlierresultsof[M.Geck,Represent.Theory4(2000)370-397]ontheHeckealgebrasoftypeDn(i.e.,oftypeG(2,2,n)).1IntroductionThetheoryofcrystal(orcanonical)basesisoneofthemostsignificantad-vancesinLietheoryoverthepasttwodecades.Itwasdiscoveredanddevel-opedbyM.Kashiwara([31])andG.Lusztig([37])around1990.Sincethen∗2000MathematicsSubjectClassification.Primary20C08,20C20,17B37.∗Keywordsandphrases.CyclotomicHeckealgebra,Fockspace,crystalbasis,Kleshchevmultipartition,Lakshmibai–Seshadripath.∗ResearchsupportedbyNationalNaturalScienceFoundationofChina(Project10401005)andbyProgramforNewCenturyExcellentTalentsinUniversityandpartlybytheURFofVictoriaUniversityofWellington.1remarkableapplicationstosomeclassicalproblemsinrepresentationtheoryhavebeenfound.Onetypicalexampleisthewell-knownLascoux–Leclerc–Thibon’sConjecture([35]),whichassertsthat,thedecompositionnumbersoftheIwahori–Heckealgebraassociatedtosymmetricgroupataprimitivee-throotofunityinC(thecomplexnumberfield)canbeobtainedfromtheevaluationat1ofthecoefficientpolynomialsofnaturalbasesappearedintheexpansionofglobalcrystalbases(i.e.,canonicalbases)ofsomeleveloneFockspacesoverthequantumaffinealgebraUq(bsle).ThisconjecturehasbeenprovedbyS.Ariki([2]),whogeneralizedittothecaseofthecyclo-tomicHeckealgebrasoftypeG(r,1,n).Asimilarconjecture([36]),whichrelatesthedecompositionnumbersoftheq-Schuralgebrawithqspecializedtoaprimitivee-throotofunityinCtoglobalcrystalbasesofFockspaceasUq(bgle)-module,hasbeenprovedbyVaragnolo–Vasserot([43]).Forfur-therexample,seetheworkofBrundan–Kleshchev([10]),wherethetheoryofcrystalbasesoftypeA(2)2lwasappliedtothemodularrepresentationsofHecke–Cliffordsuperalgebrasaswellasofdoublecoversofsymmetricgroups.Thispaperprovidesanewapplicationofthetheoryofcrystal(orcanon-ical)basestomodularrepresentationtheory.Precisely,weapplythecrystalbasistheoryforFockspacesofhigherleveloverthequantumaffinealgebraoftypeA(1)ltothemodularrepresentationsofthecyclotomicHeckealgebraH(p,p,n)oftypeG(p,p,n)inthenon-separatedcase(seeDefinition3.2).Theseparatedcasehasbeencompletelysolvedinourpreviouswork[25].Weexplicitlydescribe(intermsofcombinatoricsovercertainKleshchev’sgoodlattices)whichirreduciblerepresentationoftheAriki–KoikealgebraH(p,n)remainsirreduciblewhenrestrictedtoH(p,p,n).Thisyieldsaclas-sificationofsimplemodulesoverthecyclotomicHeckealgebraH(p,p,n)inthenon-separatedcase,generalizingourpreviouswork[24]ontheHeckealgebrasoftypeDn.Thenwegofurtherintheremainingpartofthispa-per.WeuseNaito–Sagaki’swork([40],[41])onLakshmibai–SeshadripathsfixedbydiagramautomorphismstoderiveexplicitformulaforthenumberofsimplemodulesoverthecyclotomicHeckealgebraH(p,p,n).OurformulageneralizesearlierresultofGeck[19]ontheHeckealgebraoftypeDn(i.e.,oftypeG(2,2,n)).Notethatourapproacheveninthatspecialcaseisquitedifferent,becauseitisbasedonAriki’scelebratedtheorem([2])onagener-alizationofLascoux–Leclerc–Thibon’sConjectureaswellasNaito–Sagaki’swork([40],[41])onLakshmibai–Seshadripaths,whileGeck’smethodin[19]dependsonexplicitinformationoncharactertablesandKazhdan–LusztigtheoryforIwahori–HeckealgebrasassociatedtofiniteWeylgroups—notpresentlyavailableinourgeneralG(p,p,n)cases.Asabyproduct,weget2aremarkablebijectionbetweentwosetsofKleshchevmultipartitions.Ourexplicitformulasstronglyindicatethattherearesomenewintimatecon-nectionsbetweentherepresentationofH(p,p,n)atrootsofunityandtherepresentationsofvariousAriki–Koikealgebrasofsmallersizesatvariousrootsofunity.Althoughwewillnotdiscussthesemattersinthepresentpaper,weremarkthatitseemsverylikelythedecompositionmatrixofthelattercanbenaturallyembeddedasasubmatrixofthedecompositionmatrixoftheformer.Thepaperisorganizedasfollows.Section2collectssomebasicknownresultsaboutAriki–Koikealgebras(i.e.,thecyclotomicHeckealgebrasoftypeG(r,1,n)).TheseincludeDipper–James–Mathas’sworkonthestruc-tureandrepresentationtheoryofAriki–KoikealgebrasaswellasDipper–Mathas’sMoritaequivalenceresults.ThenotionofKleshchevmultipartitionaswellasAriki’sremarkableresult(Theorem2.7)arealsointroducedthere.InSection3,wefirstrecallourpreviousworkonmodularrepresentationsofHeckealgebrasoftypeDnandoftypeG(p,p,n).Thenwegivethefirsttwomainresults(Theorem3.6andTheorem3.8)inthispaper.Theorem3.9showsthatthesetwomainresultsarevalidoveranyfieldKwhich