Some results obtained by application of the LLT algorithm

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1、REPRESENTATIONTHEORYAnElectronicJournaloftheAmericanMathematicalSocietyVolume00,Pages000{000(XxxxXX,XXXX)S1088-4165(XX)0000-0SOMERESULTSOBTAINEDBYAPPLICATIONOFTHELLTALGORITHMSINEADLYLEAbstract.ForeveryHeckealgebraoftypeA,wemaydeneadecompositionmatrix;thestruct

2、ureofeachsuchmatrixiswell{known,butingeneralthereisnowaytocomputetheentries.AnexceptionistheHeckealgebraH0=HC;!(Sn),where!isarootofunityinC.Herearecursivealgorithm,theLLTalgorithm,willproducethedecompositionmatrices{infact,theresultingmatrixprovidesa`rstapproxi

3、mation'tothedecompositionmatrixofanarbitraryHeckealgebraoftypeA.TheLLTalgorithmis,however,recursiveonn.Weshowthat,inthecaseofsomesimplepartitions,itispossibletousethealgorithmtoobtaingeneralresults;inparticular,givenaSpechtmodulecorrespondingtoapartitionwithatmo

4、stthreeparts,wewillnditscompositionfactors.Wealsogiveanindicationofthesituationinwhichthepartitioninquestionhasfourparts.1.IntroductionIn1996,arecursivealgorithmwaspublished,whichtheauthorsclaimedcouldde-terminethedecompositionmatricesoftheIwahori{HeckealgebraH

5、0=HC;!(Sn),where!wasaprimitiveethrootofunityinC.Apartfrombeinginterestingintheirownright,thedecompositionmatricesforH0willprovideinformationaboutanarbitraryHeckealgebraH=HF;q(Sn),includingthesymmetricgroupalgebraFSn.Thisclaim[4]hassincebeenproved[1],andtheresult

6、ingalgorithmisknownastheLLTalgorithm.However,itisrecursiveonn,andevenwithspeciallydesignedcomputerprograms,forexampletheGAPsharepackageSpecht[S+95,7],itisimpracticaltoobtainresultsforlargen.Nevertheless,inthecaseofsomesimplepartitions,ithasbeenpossibletoobtainex

7、plicitresults.Weconcentratehereonpartitionswithatmostthreeparts;wewilldeterminethecompositionfactorsofanySpechtmodulecorrespondingtosuchapartition.Moredetailedproofsofourresultsandfurtherinformationconcerningpartitionswithatmostfourpartsappearin[5];theargumentsa

8、ppliedaresimilartothosegiven.Webeginbyestablishingsomenotation.Wexanintegere2.Throughout,H0willdenotetheIwahori{HeckealgebraHC;!(Sn)where!isaprimitiveethrootofunity