Extensions of modules over Schur algebras, symmetr

EXTENSIONSOFMODULESOVERSCHURALGEBRAS,SYMMETRICGROUPSANDHECKEALGEBRASSTEPHENR.DOTY,KARINERDMANN,ANDDANIELK.NAKANOAbstract.Westudytherelationbetweenthecohomologyofgen-erallinearandsymmetricgroupsandtheirrespectivequantiza-tions,usingSchuralgebrasandstandardhomologicaltechniquestobuildappropriatespectralsequences.Asourmethodstinsideamuchmoregeneralcontextwithinthetheoryofnite-dimensionalalgebras,wedevelopourresultsrstinthatgeneralsetting,andthenspecializetotheabovesituations.Fromthisweobtainnewproofsofseveralknownresultsinmodularrepresentationtheoryofsymmetricgroups.Moreover,wereducecertainquestionsaboutcomputingextensionsforsymmetricgroupsandHeckealgebrastoquestionsaboutextensionsforgenerallineargroupsandtheirquantizations.1.Introduction1.1.Webeginbyexplainingourmotivation.LetEbethevectorspaceofn-dimensionalcolumnvectorsoveraeldk.Weassumethatkisinnitejustforthisintroduction.ThegenerallineargroupGLnoverkactsnaturallyonEbymatrixmultiplication,andthisactionextendstoanactiononthetensorpowerEd,byactingthesameineachtensorposition.Thisactioncommuteswiththeactionofthesymmetricgroupd,actingbyplace-permutation.Butmuchmoreistrue.LetkGLn!End(Ed)kdbethecorrespondingrepresentations,extendedbylinearitytoeachgroupalgebra.Thentheimageofeachrepresentationisthecompletecentralizeroftheother,i.e.im=Endd(Ed);im=EndGLn(Ed):Thisdouble-centralizertheorem,whichgoesbacktoSchur[S2]inzerocharacteristicandDeConciniandProcesi[DP]ingeneral,hashadaDate:June2000.ThethirdauthorwaspartiallysupportedbyNSFgrantDMS-9800960.12STEPHENR.DOTY,KARINERDMANN,ANDDANIELK.NAKANOstrongimpactonthedeepinteractionsbetweentherepresentationthe-oriesofgenerallinearandsymmetricgroups,andisinsomesenseourstartingpoint.Amaingoalofthispaperistoexploittheabovesetuptorelatethecohomologyofthetwogroups.WedothisbymeansoftheSchuralgebraS(n;d),whichcanbedenedastheimageofthemapabove.Themodulesforthisnite-dimensionalalgebraarepreciselytheGLn-moduleswhosecoecientsdependonthematrixcoordinatefunctionsofGLnashomogeneouspolynomialsoftotaldegreed.ManyinterestingmodulesforGLn(e.g.Weylmodules,simplemodules)arehomogeneousorarecloselyrelatedtohomogeneousmodules.Whennissucientlylarge(nd)themoduleEdisaprojectivesummandofthealgebraA=S(n;d),i.e.thereisanidempotente2AsuchthatAe=Ed[Gr1,(6.4f)].ItfollowsimmediatelyfromthedoublecentralizerpropertythateAeisisomorphicwithkd.ThusonehasanexactcovariantfunctorF(sometimescalledthe\Schurfunctor)goingfromA-modtoeAe-mod,whichisgivenbyM7!eM.EversincethepublicationofGreen’smonograph[Gr1],thisfunctorhasbeenexploitedtorelatethemodularrepresentationtheoryofGLntothatofd;itsuseincharacteristiczerogoesallthewaybacktoSchur’sthesis[S1].Everythingwehavesaidthusfaradmitsofquantization,andindeedwearealsointerestedinstudyingtherelationbetweenthecohomologyofquantumgenerallineargroupsandHeckealgebrasinTypeA.HeretheSchuralgebramustbereplacedbyitsquantization,theq-SchuralgebraintroducedbyDipperandJames[DJ1,DJ2].Thuswearriveatourgeneralcontext:wehaveanidempotente6=1inanarbitrarynite-dimensionalalgebraA,andwewanttorelateextensionsinA-modtoextensionsineAe-mod.WedothisbymeansofthetwonaturaladjointsGHom,GtotheSchurfunctorFarisingfromtheidenticationseM=HomA(Ae;M)=eAAMandtheassociatedGrothendieckspectralsequenceswhichareimplicitintheinteractionsofthesefunctors.Wethenobtainseveralapplica-tions,inseveralcasesobtainingnewproofsofresultsobtainedearlierbymoreadhocmethods.Inasense,itseemsthatthesestandardhomologicalmethodshavebeenoverlookedintheliteratureconcernedwiththerelationbetweensymmetricandgenerallineargroups,sooneEXTENSIONSOFMODULES3ofourpurposesistopointouttheapplicabilityofthesetoolstosuchquestions.Anearlierpreprintversionofthispaperhasbeencirculatedforsometime,andnowthereareseveralpaperswhichrelyontechniquesandreductionsfromthisonetoobtainnewresultsaboutsymmetricgroupcohomology.In[KN],extensionsfortheSchuralgebrasarecomparedwithcorrespondingextensionsforthesymmetricgroupinacertainexplicitrangeofdegrees.ThestabilityresultsobtainedthereprovideaeasyproofofaconjecturemadebyBurichenko,KleshchevandMartin[BKM]onthecohomologyofdualSpechtmodules.Otherapplicationsaregivenin[DN].Weshouldaddthattheuseofquasi-hereditaryalgebrastocomputegroupcohomologyalsoappearsinrecentworkofBendel,Pillenandthesecondauthor[BNP].Inthatpaper,itisshownthatthattherearespectralsequenceswhichstartwithmodulesinaboundedcategoryofG-moduleswhereGisareductivealgebraicgroupG,andconvergetotheextensionsovertheniteChevalleygroupG(Fq).1.2.Summary.Thepaperisorganizedasfollows.Inthenextsectionweconsiderthegeneralsituationofanarbitrarynite-dimensionalalge-braAandidempotente2A.WeconstructtheGrothendieckspectralsequencesmentionedabove.Theresultingve-termexactsequencesgiverelationshipsbetweenExt1-groupsinthecategoriesofA-modulesandeAe-modules.Inparticular,todescribeExt1inthecategoryofeAe-modules,oneneedstocomputeimagesundertheadjointsGHomandGoftheSchurfunctorF.Inthethirdsectionweinvestigatepropertiesoftheseadjointfunc-tors.WegiveadescriptionoftheimageG(N)foraneAe-moduleNwhereG=GHomo