On the rank of a Coxeter group

arXiv:0706.3911v1[math.GR]26Jun2007OntheRankofaCoxeterGroupMichaelL.MihalikandJohnG.RatcliffeMathematicsDepartment,VanderbiltUniversity,NashvilleTN37240,USA1IntroductionLetWbeaCoxetergroupwithCoxetergeneratorsS.TherankoftheCoxetersystem(W,S)isthecardinality|S|ofS.TheCoxetersystem(W,S)hasfiniterankifandonlyifWisfinitelygeneratedbyTheorem2(iii),Ch.IV,§1of[1].If(W,S)hasinfiniterank,then|S|=|W|,sinceeveryelementofWisrepresentedbyafiniteproductofelementsofS.ThusifWisnotfinitelygenerated,therankof(W,S)isuniquelydeterminedbyW.IfWisfinitelygenerated,thenWmayhavesetsofCoxetergeneratorsSandS′ofdifferentranks.Inthispaper,wedeterminethesetofallpossibleranksforanarbitraryfinitelygeneratedCoxetergroupW.ThispaperisacontinuationofourpreviouspaperwithStevenTschantz[6]inwhichwestudiedtherelationshipbetweentwosetsSandS′ofCoxetergeneratorsofafinitelygeneratedCoxetergroupW.AbasicsubsetofSisamaximalsubsetBofSsuchthatBgeneratesanirreducible,noncyclic,finitesubgroupofW.In[6],weprovedtheBasicMatchingTheoremwhichsaysthatthereisanaturalbijection(matching)betweenthebasicsubsetsofSandthebasicsubsetsofS′.AbasicsubsetBofSmatchesabasicsubsetB′ofS′ifandonlyif[hBi,hBi]isconjugateto[hB′i,hB′i]inW.Usuallymatchingbasicsubsetsgenerateisomorphicgroups,inwhichcase,wesaythatthebasicsubsetsmatchisomorphically;however,thereareexceptions,duetowellknownisomorphismsbetweenirreducibleandreduciblefiniteCoxetergroups(forinstancethedihedralgroupD2(6)oforder12andA1×A2).WeshowedthatnonisomorphicmatchingofbasicsubsetscanbeunderstoodbyblowingupCoxetergeneratingsets.Thisisaproceduretoreplacea1givenCoxetergeneratingsetSbyaCoxetergeneratingsetRsuchthat|R|=|S|+1.In[6],weprovedthatthereexistsasetofCoxetergeneratorsS′ofWsuchthatabasicsubsetBofSmatchesabasicsubsetB′ofS′with|hBi||hB′i|ifandonlyifScanbeblownup.WeprovedthatShasmaximumrankoverallsetsofCoxetergeneratorsofWifandonlyifScannotbeblownup.Inthispaper,westudythereverseprocedureofblowingdownCoxetergeneratingsets,whichwasintroducedbyMihalikin[5].Wefirstdeterminenecessaryandsufficientconditionson(W,S)suchthatthereexistsasetofCoxetergeneratorsS′ofWsuchthatabasicsubsetBofSmatchesabasicsubsetB′ofS′with|hBi||hB′i|.Wethendeterminenecessaryandsufficientconditionson(W,S)suchthatWhasasetofCoxetergeneratorsS′suchthat|S′||S|.Asanapplication,wedeterminetherankspectrumofW.2PreliminariesACoxetermatrixisasymmetricmatrixM=(m(s,t))s,t∈Swithm(s,t)eitherapositiveintegerorinfinityandm(s,t)=1ifandonlyifs=t.ACoxetersystemwithCoxetermatrixM=(m(s,t))s,t∈Sisapair(W,S)consistingofagroupWandasetofgeneratorsSforWsuchthatWhasthepresentationW=hS|(st)m(s,t):s,t∈Sandm(s,t)∞iIf(W,S)isaCoxetersystemwithCoxetermatrixM=(m(s,t))s,t∈S,thentheorderofstism(s,t)foreachs,tinS,andsoaCoxetersystemdeter-minesitsCoxetermatrix;moreover,anyCoxetermatrixM=(m(s,t))s,t∈SdeterminesaCoxetersystem(W,S)whereWisdefinedbytheabovepre-sentation.If(W,S)isaCoxetersystem,thenWiscalledaCoxetergroupandSiscalledasetofCoxetergeneratorsforW,andthecardinalityofSiscalledtherankof(W,S).Let(W,S)beaCoxetersystem.TheCoxeterdiagram(C-diagram)of(W,S)isthelabeledundirectedgraphΔ(W,S)withverticesSandedges{(s,t):s,t∈Sandm(s,t)2}suchthatanedge(s,t)islabeledbym(s,t).ACoxetersystem(W,S)issaidtobeirreducibleifitsC-diagramisconnected.2Avisiblesubgroupof(W,S)isasubgroupofWoftheformhAiforsomeA⊆S.AvisiblesubgrouphAiof(W,S)issaidtobeirreducibleif(hAi,A)isirreducible.AsubsetAofSissaidtobeirreducibleifhAiisirreducible.AsubsetAofSissaidtobeacomponentofSifAisamaximalirreduciblesubsetofSorequivalentlyifΔ(hAi,A)isaconnectedcomponentofΔ(W,S).Thepresentationdiagram(P-diagram)of(W,S)isthelabeledundirectedgraphΓ(W,S)withverticesSandedges{(s,t):s,t∈Sandm(s,t)∞}suchthatanedge(s,t)islabeledbym(s,t).Wecontinuewiththeterminologyof[6].Inparticular,weuseCoxeter’snotationonp.297of[2]fortheirreduciblesphericalsimplexreflectiongroupsexceptthatwedenotethedihedralgroupDk2byD2(k).SubscriptsdenotetherankofaCoxetersysteminCoxeter’snotation.Coxeter’snotationpartlyagreeswithbutdiffersfromBourbaki’snotationonp.193of[1].CoxeterprovedthateveryfiniteirreducibleCoxetersystemisisomorphictoexactlyoneoftheCoxetersystemsAn,n≥1,Bn,n≥4,Cn,n≥2,D2(k),k≥5,E6,E7,E8,F4,G3,G4.See§3of[6]fordefinitions.Foruniformityofnotation,wedefineB3=A3,D2(3)=A2andD2(4)=C2.Let(W,S)beaCoxetersystem.AbasicsubsetofSisamaximalirre-duciblesubsetBofSsuchthathBiisanoncyclicfinitegroup.IfBisabasicsubsetofS,wecallBabaseof(W,S)andhBiabasicsubgroupofW.Theorem2.1(BasicMatchingTheorem,Theorem4.18[6])LetWbeafinitelygeneratedCoxetergroupwithtwosetsofCoxetergeneratorsSandS′.LetBbeabaseof(W,S).ThenthereisauniqueirreduciblesubsetB′ofS′suchthat[hBi,hBi]isconjugateto[hB′i,hB′i]inW.Moreover,1.thesetB′isabaseof(W,S′),andwesaythatBandB′match,2.if|hBi|=|hB′i|,thenBandB′havethesametypeandthereisanisomorphismφ:hBi→hB′ithatrestrictstoconjugationon[hBi,hBi]byanelementofW,andwesaythatBandB′matchisomorphically,3.if|hBi||hB′i|,theneitherBhastypeB2q+1andB′hastypeC2q+1forsomeq≥1orBhastypeD2(2q+1)andB′hastypeD2(4q+2)forsomeq≥1.Moreover,thereisamonomorphismφ:hBi→hB′ithatrestrictstoconjugationon[hBi,hBi]byanelem