Decomposition of tensor products of modular irredu

DeompositionoftensorprodutsofmodularirreduiblesforSL2StephenDotyandAnneHenkeMarh20,2002AbstratWeusetiltingmodulestostudythestrutureofthetensorprodutoftwosimplemodulesforthealgebraigroupSL2,inpositivehara-teristi,obtainingatwistedtensorproduttheoremforitsindeom-posablediretsummands.Variousotherrelatedresultsareobtained,andnumerousexamplesareomputed.IntrodutionWestudythestrutureofLL0whereL;L0aresimplemodulesforthealgebraigroupSL2=SL2(k)overanalgebraiallylosedeldkofposi-tiveharateristip.Thesolutiontothisproblemiswell-knownandeasilyobtainedinharateristizero,butinpositiveharateristitheproblemissigniant.GivenLL0,theinitialquestionistodesribeitsindeomposablediretsummands.ThisisansweredinTheorem2.1.Itturnsoutthateahsuhdiretsummandisexpressibleasatwistedtensorprodutofertain\smallindeomposabletiltingmoduleswherethestrutureofthelatterisompletelyunderstood(seeLemma1.1).Wenotethatforp=2themoduleLL0isalwaysindeomposable,inontrasttowhathappensforpodd.Theindeomposablesummandsthemselvesarealwaysontravariantlyself-dual,withsimplesole(andhead),andtheyourassubquotientsoftiltingmodules(seeTheorem2.7).Ontheotherhand,eahtiltingmoduleours1asadiretsummandofsomeLL0(seeTheorem2.6).TheseresultsprovidethestartingpointforalulatingtheexamplesinSetion5and6.Infat,wesuggestthatthereaderstartbybrowsingthroughtheexamplesinSetions5and6.Theloserelationshipbetweentensorprodutsofsimplesandtiltingmoduleswillbeapparentfromtheseexamples.Sineexamplesoftiltingmodulestruturearerareintheliterature,theseomputationsshouldbeofindependentinterest.OurgeneralresultsaregiveninSetions2through4.Inpartiular,welas-sifypreiselywhihindeomposablesummandsofLL0aretilting,andweobtainaresultexpressingertainindeomposabletiltingmodulesastensorprodutsoftwosimplemodules(usuallyinmorethanoneway).InSetions3and4westudyindetailthetensorprodutsLL(1)whereLisarbitrary.Hereweobtainalassofuniserialandbiserialtiltingmodulesandtheir\shifts.ThemethodsusedhereanbeappliedtoobtainthestrutureofalltensorprodutsLL0whereL0=L(a)withap1,althoughweformulatethepreisestatementonlyforLL(2).TheresultsofthepaperarerelatedtoresultsofAlperin[A1℄,BrundanandKleshhev[BK℄,andErdmannandHenke[EH1,EH2℄.OurmaintehnialtoolsareSteinberg’stensorproduttheoremandDonkin’sinterpretation[Do℄,intheontextofalgebraigroups,ofRingel’stheory[R℄oftiltingmodules.1PreliminariesThesetX=X(T)ofweightsforamaximaltorusTinthealgebraigroupSL2willbeidentiedwiththesetZofintegers,asusual.Thendominantweightsorrespondtononnegativeintegers.IfrissuhthenwewriteL(r)forthesimpleSL2-moduleofhighestweightrand(r)fortheWeylmoduleofthatsamehighestweight.Wewriter(r)forthetranspose(ontravariant)dualof(r).ByatheoremofRingel(see[R℄),thereexistsauniqueindeom-posablemoduleT(r)ofhighestweightrsuhthatT(r)hasboth-ltrationandr-ltration.ThemodulesT(r)arethe(partial)tiltingmodules.IfamoduleMhasaompositionseries0=M0M1Mk=MwithsimplefatorsSi=Mi=Mi1fori=1;:::;kthenwedenotethatomposition2seriesbywriting[S1;S2;:::;Sk℄:Webeginwithsomeeasylemmasontiltingmodules.Ourrstresultde-sribesthemodulestrutureofertainsmalltiltingmodules,whihwillturnouttobethebasimaterialoutofwhihalltiltingmodulesandtensorprodutsofsimplesarebuiltup.Lemma1.1(a)For0up1wehaveT(u)=L(u)=r(u)=(u).(b)Forpu2p2themoduleT(u)isuniserialanditsuniqueompo-sitionserieshastheform[L(2p2u);L(u);L(2p2u)℄.Moreover,T(u)isanon-splitextensionof(2p2u)by(u)(or,dually,ofr(u)byr(2p2u)).Proof.Reallthatr(r)=Sr(E),therthsymmetripowerofthenaturalmoduleE.Part(a)followsimmediatelyfromthefatthatr(u)=(u)=L(u)issimplefor0up1.Thisfatiswell-knownandfollowsforinstanefromthestronglinkagepriniple,orfromknownresults[D1℄onthestrutureofSr(E).Part(b)isaspeialaseof[EH2,Proposition2.3℄.Or:WriteG=SL2andsetG1=KerF(theFrobeniuskernel).AsobservedbyDonkin[Do,x2,Example1℄,for0p1wehaveanisomorphismT(2p2)=Q(),whereQ()istheunique(uptoG-isomorphism)G-modulesuhthatQ()jG1isisomorphiwiththeprojetiveoverofL()jG1.TheexisteneofthisG-moduleliftfollowsfromresultsofJantzen,extendingearlierresultsofBallard.Nowweanapply[J,II,11.4Prop.℄toomputetheformalharaterofQ().Settingu=2p2andrestritingto0p2,weanapplytheaformentioneddesriptionofther’sassymmetripowerstoobtainpart(b)ofthelemma.Letusallthetiltingmodulesdesribedinthepreedingresultfundamental.Asweshallsee(inLemma1.4ahead)anytiltingmoduleforSL2anbeexpressedasatwistedtensorprodutoffundamentaltiltingmodules.More-over,inTheorem2.1weshallseethattheindeomposablesummandsofLL0analsobeexpressedassuhatwistedtensorprodut.(Theindeompos-ablesummandsarenotneessarilytilting,however.)For0u2p2,we3denotebyeuthehighestweightofthesole(andhead)ofT(u),sothateu=(uifup1,2p2uotherwise:(1.2)Mostasesofthenextlemmaappearalreadyin[EH1,Lemma4℄.Lemma1.3LetL;L0betwosimplemodulesinthebottomalove,i.e.theirhighestweightsareinlusivelybetween0andp1.ThenLL0istilting,andisomorphiwiththediretsumofT(u)asuvariesoverasetW(L;L0)ofweightswhihanbeomputedasfollows.Letr(resp.,s)bethelarger(resp.,smaller)ofthehighestweightsofL;L0.Listtheweightsr+s,r+s2;:::;rs.Foreahuponthislist,stri