On bases of centres of Iwahori-Hecke algebras of t

arXiv:math/0401036v3[math.QA]2May2005ONBASESOFCENTRESOFIWAHORI–HECKEALGEBRASOFTHESYMMETRICGROUPANDREWFRANCISANDLENNYJONESAbstract.In1990,usingnorms,thesecondauthorconstructedabasisforthecentreoftheHeckealgebraofthesymmetricgroupSnoverQ[ξ][13].Anintegral“minimal”basiswaslatergivenbythefirstauthorin1999[5],following[9].Inprincipleonecanthenwriteelementsofthenormbasisasintegrallinearcombinationsofminimalbasiselements.Inthispaperwefindanexplicitnon-recursiveexpressionforthecoefficientsappearingintheselinearcombinations.Thesecoeffi-cientsareexpressedintermsofcertainpermutationcharactersofSn.Intheprocessofestablishingthismaintheorem,weprovethefollowingitemsofindependentinterest:aresultontheprojectionofthenormsontoparabolicsubalgebras,theexistenceofaninnerproductontheHeckealgebrawithsomeinterestingproperties,andtheexistenceofapartialorderingonthenorms.0.IntroductionTherearenowthreedistinctdescriptionsofthecentreoftheIwahori–HeckealgebraHofthesymmetricgroupSn.Ithastwonicebases,oneconsistingofnormsoverQ[ξ][13],andonea“minimalbasis”ofclasselementsoverZ[ξ][9,5].Thirdly,itisnowknownthatthesymmetricfunctionsinMurphyelementsarepreciselythecentreofHoverZ[ξ][10],anditfollowsthattheelementarysymmetricfunctionsinMurphyelementsgeneratethecentreoverZ[ξ].Anaturalquestionisthentoask“Howarethesedescriptionsrelated?”.TherelationshipbetweentheelementarysymmetricfunctionsofMurphyelementsandthemin-imalbasisisnowknownpreciselyatleastinonedirection[4],butrelationshipswiththenormbasishavebeenopaque.Furthermore,theelucidationoftheconnectionsbetweenthenormbasisandtheotherbasesisofinterestsincethenormsof[13]arenaturalcentralstructuresDate:February1,2008.2000MathematicsSubjectClassification.Primary20C08.Keywordsandphrases.Heckealgebra,center,minimalbasis,norm.12ANDREWFRANCISANDLENNYJONESwhichhavebeenusedtodefineBrauer-typehomomorphismsforHeckealgebras[12,3,6]andq–Schuralgebras[2].ThegoalofthispaperistodescribeanexplicitrelationshipbetweenthenormbasisandtheminimalbasisforthecentreoftheHeckealgebraofthesymmetricgroupSn.Thisrelationshipisgivenbyanexpressionforthecoefficientsofclasselements(theminimalbasis)astheyappearinthenorms.ThesecoefficientsaredescribedintermsofthevaluesofcertainpermutationcharactersofSn.Letαandλbepartitionsofn,withwαanelementoftheconjugacyclassCαofSn.LetlλandlαbethelengthsoftheminimalelementsinthecorrespondingconjugacyclassesofSn,andletξbethedefiningindeterminateoftheHeckealgebra.Let(1Sλ)SnbethepermutationcharacterofSnwhicharisesfromtheinductiontoSnofthetrivialcharacterontheparabolicsubgroupSλ.Themainresultisasfollows.Theorem9.2:LetbαbeanelementofthenormbasisandletΓλbeanelementoftheminimalbasis.Thenbα=Xλ⊢n(1Sλ)Sn(wα)ξlλ−lαΓλ.Aconsiderableamountofmachinery,whichinvolvesseveralresultsofindependentinterest,isdevelopedinthecourseofobtainingTheorem9.2.ThepreliminarySection1introducesmostofthebasicdefinitionsandnotationusedthroughoutthepaper.Thereadermaywishtoskimthissectionandreturnforreferenceasrequiredlaterinthepaper.Sec-tion2containsresultsaboutdoublecosetrepresentativesofparabolicsubgroupsinthesymmetricgroupSnwhicharerequiredforSection8.AformulaforthesquareoftheHeckealgebraelementcorrespondingtoadistinguisheddoublecosetrepresentativeisgiveninSection3.InSection4,themainpropertiesofthebasesforthecentrearebrieflyreprised.Section5introducesaninnerproductontheHeckealgebraandgivessomeelementaryproperties.InSection6,wefindthecoef-ficientofΓλinbαwhenαistrivial(Theorem6.3),whileinSection7wedeterminethecoefficientoftheCoxeterclasselementΓ(n)inbαforallα⊢n(Theorem7.4).ToestablishTheorem7.4,weshowthatthebasisofnormssatisfiesapartialorderconsistentwiththerefinementorderonpartitions(Theorem7.2).ThedescriptionsofcoefficientsinTheorems6.3and7.4arelatermaderedundantbyTheorem9.2,butarenecessaryforitsproof.Section8givesthemainprojectiontheorem(Theorem8.1),whichusesaMackey-typedecompositiontogivearuleforprojectingnormsontoamaximalparabolicsubalgebra.ThisresultCENTRESOFHECKEALGEBRAS3hasbeenusedtostudytheBrauerhomomorphismin[3].InSection9,Theorem8.1isgeneralizedtoaruleforprojectingontoarbitrarypara-bolicsubalgebras(Theorem9.1),andthemaintheoremquotedaboveisdeduced.Finally,themainresultisdemonstratedinSection10withsomeexamples.Theauthorsthanktherefereeformanyvaluablecommentsandsug-gestions.1.DefinitionsandnotationThroughoutwetakeNtomeanthesetofnon-negativeintegers.1.1.Compositions,partitionsandmultipartitions.Acomposi-tionλisafiniteorderedsetofpositiveintegers.Ifλ=(λ1,...,λr),theλiarecalledthecomponentsofλ.Ifλisacompositionwewrite|λ|=Pri=1λi.If|λ|=nwesayλisacompositionofn,andwewriteλn.Twocompositionsaresaidtobeconjugateiftheyhavethesamecomponents.Ifλ=(λ1,...,λr)nthenwedefineλ−1tobethecompositionofnobtainedfromλbyreplacingeachλi1bythejuxtaposedorderedpairofpositiveintegersλi−1and1.Forexample,ifλ=(3,4,1,7)thenλ−1=(2,1,3,1,1,6,1).Ifλandμarecompositionsofnandeitherλ=μorλcanbeobtainedfromμbyaddingtogetheradjacentcomponentsofμ,wesayμisarefinementofλandwriteμ≤λ.Apartitionofnisacompositionwhosecomponentsareweaklyde-creasingfromlefttoright.Ifλisapartitionofnwewriteλ⊢n.Amultipar