Representation type of Schur superalgebras

arXiv:math/0411390v1[math.RT]17Nov2004REPRESENTATIONTYPEOFSCHURSUPERALGEBRASDAVIDJ.HEMMER,JONATHANKUJAWA,ANDDANIELK.NAKANOAbstract.LetS(m|n,d)betheSchursuperalgebrawhosesupermodulescorrespondtothepolynomialrepresentationsofthesupergroupGL(m|n)ofdegreed.Inthispaperwedeterminetherepresentationtypeofthesealgebras(i.e.classifytheoneswhicharesemisimple,havefinite,tameandwildrepresentationtype).Moreover,weprovethatthesealgebrasareingeneralnotquasi-hereditaryandhaveinfiniteglobaldimension.1Introduction1.1Acentralprobleminthestudyoffinite-dimensionalalgebrasistounderstandthestructureoftheindecomposablemodules.Asafirststeponewouldliketoknowhowmanyindecomposablemodulesagivenalgebraadmits.Anyfinite-dimensionalalgebracanbeclassifiedintooneofthreecategories:finite,tame,orwildrepresentationtype.Afinite-dimensionalalgebraAhasfiniterepresentationtypeifAhasfinitelymanyindecomposablemodulesuptoisomorphism.IfAisnotoffinitetypethenAisofinfiniterepresentationtype.Algebrasofinfiniterepresentationtypeareeitheroftamerepresentationtypeorofwildrepresentationtype.Foralgebrasoftamerepresentationtypeonehasachanceofclassifyingalltheindecomposablemodulesuptoisomorphism.Therehavebeenmuchprogressinthepasttenyearsindeterminingtherepresentationtypeofimportantclassesoffinite-dimensionalalgebras.TheSchuralgebrasandtheq-Schuralgebraarefinite-dimensionalalgebraswhichariseintherepresentationtheoryofthegenerallineargroupsandsymmetricgroups.AcompleteclassificationoftherepresentationtypeofthesealgebraswasgivenbyDoty,Erdmann,MartinandNakano[E,DN,DEMN,EN1].Thesealgebrasarefundamentalexamplesofquasi-hereditaryalgebras(orequivalentlyhigh-estweightcategories)whichwereintroducedbyCline,ParshallandScott[CPS].TheblocksfortheordinaryandparabolicBGGcategoryOforfinite-dimensionalcomplexsemisimpleLiealgebrasareotherimportantexamplesofhighestweightcategories.ResultspertainingtotherepresentationtypefortheseblockswereobtainedindependentlybyFutorny,NakanoandPollack[FNP],andBr¨ustle,K¨onigandMazorchuk[BKM]fortheordinarycategoryDate:October2004.1991MathematicsSubjectClassification.Primary20C30.ResearchofthefirstauthorwassupportedinpartbyNSFgrantDMS-0102019.ResearchofthesecondauthorwassupportedinpartbyNSFgrantDMS-0402916.ResearchofthethirdauthorwassupportedinpartbyNSFgrantDMS-0400548.12DAVIDJ.HEMMER,JONATHANKUJAWA,ANDDANIELK.NAKANOOandmorerecentlyfortheparaboliccategoryObyBoeandNakano[BN].Forquasi-hereditaryalgebrastheprojectivemodulesadmitfiltrationsbycertainstandardmodules.Thisinformationallowsonetodeterminethestructureoftheprojectivemoduleswhichinturncanleadtoanexpressionofthealgebraviaquiverandrelations.Inmanycasesthisinformationcanbeusedtodeterminetherepresentationtypeofthealgebra.Otherimportantclassesoffinite-dimensionalalgebrasarequasi-Frobeniusalgebraswhereallprojectivemodulesareinjective.Thesealgebrasallhaveinfiniteglobaldimension(ex-ceptwhenthealgebraissemisimple).Forgroupalgebras,restrictedenvelopingalgebras,andFrobeniuskernelshomologicalinformationinvolvingthetheoryofcomplexityandsup-portvarietiescanbeusedtodeduceinformationabouttherepresentationtypeofthesealgebras.ForHeckealgebrasoftypeA,therepresentationtypeoftheblockswasgivenbyErdmannandNakanoin[EN2].ArikiandMathas[AM]classifiedtherepresentationtypeforHeckealgebrasoftypeBusingFockspacemethods.Ariki[A]recentlyhasextendedthisclassificationtoHeckealgebrasofclassicaltype.ThemainresultofthispaperisacompleteclassificationoftherepresentationtypefortheSchursuperalgebrasS(m|n,d).ThesealgebrasareofparticularinterestbecausethecategoryofS(m|n,d)-supermodulesisequivalenttothecategoryofpolynomialrepresen-tationsofdegreedforthesupergroupGL(m|n).Also,aswiththeclassicalSchuralgebra,thereisaSchur-WeyldualitybetweenS(m|n,d)andthesymmetricgroupΣd(see[BK]).WeshallshowthatinmostcasesS(m|n,d)isnotquasi-hereditary,whichiscontrarytoarecentconjecturegivenbyMarkoandZubkov.SincetheSchursuperalgebrasarenotquasi-hereditarytheaforementionedfiltrationtechniquescannotbeused.Inordertoob-taininformationaboutthebasicalgebraforcertainSchursuperalgebraswewillcomputetheendomorphismalgebrasofdirectsumsofsignedYoungmodules.Thiswillentailknow-ingthestructuresofcertainsignedYoungmodules,whichisofindependentinterestfromtheviewpointoftherepresentationtheoryofsymmetricgroups.1.2WereviewbelowtheresultsfortheordinarySchuralgebraS(m,d),althoughtheproofforS(m|n,d)doesnotrelyontheseresults.TherepresentationtypeofS(m,d)wasdeterminedin[E,DN,DEMN]1.(1.2.1)Theorem.LetS(m,d)betheSchuralgebraoverkwherechark≥0.(a)S(m,d)issemisimpleifandonlyifoneofthefollowingholds(i)chark=0;(ii)dp;(iii)p=2,m=2,d=3.(b)S(m,d)hasfiniterepresentationtypeifandonlyifoneofthefollowingholds(i)p≥2,m≥3,d2p;(ii)p≥2,m=2,dp2;(iii)p=2,m=2,d=5,7.(c)S(m,d)hastamerepresentationtypeifandonlyifoneofthefollowingholds(i)p=3,m=3,d=7,8;1ThestatementthatS(2,11)forp=2hastamerepresentationtypewasinadvertentlyomittedfromtheoriginalstatementofthetheorem[DEMN,Thm.1.2(A)].WethankAlisonParkerforpointingthisouttous.REPRESENTATIONTYPEOFS(m|n,d)3(ii)p=3,m=2,d=9,10,11;(iii)p=2,m=2,d=4,9,11.(d)InallothercasesnotlistedaboveS(m,d)haswild