Crystals for Demazure Modules of Classical Affine

arXiv:q-alg/9707014v111Jul1997CrystalsforDemazureModulesofClassicalAffineLieAlgebrasAtsuoKuniba∗,KailashC.Misra†,MasatoOkado‡,TaichiroTakagi§andJunUchiyama¶AbstractWestudy,inthepathrealization,crystalsforDemazuremodulesofaffineLiealgebrasoftypesA(1)n,B(1)n,C(1)n,D(1)n,A(2)2n−1,A(2)2n,andD(2)n+1.WefindaspecialsequenceofaffineWeylgroupelementsfortheselectedperfectcrystal,andshowifthehighestweightislΛ0,theDemazurecrystalhasaremarkablysimplestructure.0IntroductionLetgbeasymmetrizableKac-Moodyalgebra.LetUq(g)beitsquantizeduniver-salenvelopingalgebraandV(λ)betheintegrableUq(g)-modulewithdominantintegralhighestweightλ.LetWbetheWeylgroupofg.FixinganelementwofW,theDemazuremoduleVw(λ)isdefinedasafinitedimensionalsubspaceofV(λ)generatedfromtheextremalweightspaceV(λ)wλbytheeigeneratorsofUq(g).ThoughtheDemazuremoduleitselfcanbedefinedinthesamewayfortheclassicalcaseq=1,westayinthequantumcase.Thereasonistheexistenceofa“good”basis.Let(L(λ),B(λ))bethecrystalbaseofV(λ)[K1].In[K2],KashiwarashowedthereexistsasubsetBw(λ)ofB(λ)suchthatVw(λ)=Mb∈Bw(λ)Q(q)Gλ(b).∗InstituteofPhysics,UniversityofTokyo,Komaba,Tokyo153,Japan†DepartmentofMathematics,NorthCarolinaStateUniversity,Raleigh,NC27695-8205,USA‡DepartmentofMathematicalScience,FacultyofEngineeringScience,OsakaUniversity,Toyonaka,Osaka560,Japan§DepartmentofMathematicsandPhysics,NationalDefenseAcademy,Yokosuka239,Japan¶DepartmentofPhysics,RikkyoUniversity,Nishi-Ikebukuro,Tokyo171,Japan1HereGλ(b)isthelowerglobalbase[K3].Moreover,Bw(λ)hasaquiteremark-ablesimplerecursiveproperty:Ifriw≻w,thenBriw(λ)=[n≥0˜fniBw(λ)\{0}.HereriisasimplereflectionofWand≻denotestheBruhatorder.Infact,usingthispropertyKashiwaragaveanewproofofDemazure’scharacterformulaforanarbitrarysymmetrizableKac-Moodyalgebra[K2].LetusnowfocusonthequantumaffinealgebraUq(g),wheregisofaffinetype.Inthiscase,wehaveadescriptionofB(λ)intermsofpaths[KMN1,KMN2].Roughlyspeaking,thesetofpathsisasuitablesubsetofthehalfinfinitetensorproductofa“perfect”crystalBwhichisacrystalofafinitedimensionalU′q(g)-modulehavingsomeniceproperties[KMN1].Onthisset,theactionsof˜eiand˜fiareexplicitlygiven.In[KMOU],wegaveacriterionforBw(λ)tohaveatensorproductstructure.Todescribethegeneralsituation,themixingindexκwasintroduced.Takingκ=1forsimplicity,theresultin[KMOU]isstatedasfollows.Consideranincreasingsequence{w(k)}k≥0ofWwithrespecttotheBruhatorder.IfaperfectcrystalBand{w(k)}satisfyseveralassumptions,thenBw(k)(λ)isgivenbyB(λ)⊂···⊗B⊗B⊗B⊗B⊗···⊗B∪Bw(k)(λ)=···⊗bj+2⊗bj+1⊗B(j)a⊗B⊗···⊗B.(0.1)Herej,aaredeterminedfromk,B(j)aisasubsetofB,andb=···⊗bj⊗···⊗b1isthegroundstatepathcorrespondingtothehighestweightvectorinB(λ).Thepurposeofthisarticleistoshowthatifλ=lΛ0(wealsodiscusssomeothersimilarcases),wecanfindthesequence{w(k)}satisfying(0.1)forgofclassicaltypes(i.e.g=A(1)n,B(1)n,D(1)n,A(2)2n−1,A(2)2n,D(2)n+1,C(1)n).WechoosetheperfectcrystalBfromthelistin[KMN2]exceptfortheC(1)ncase.Toillustrate,wetakeanexampleofg=A(1)3,B=B(Λ2).ThecrystalgraphofBisgiveninFigure1.SinceBisalevel1perfectcrystal,letustakeλ=Λ0.Thenthepathbisgivenbyb=···⊗12⊗34⊗12⊗34.Inthiscase,thesequenceofWeylgroupelements{w(k)}k≥0isgivenasfollows:w(0)=1,w(k+1)=riw(k)(k≥0),wherei=0(k≡0,3),i=3(k≡1,6),i=1(k≡2,5),i=2(k≡4,7).Here≡denotes‘congruencemodulo8’.Theintegersj,ain(0.1)aredeterminedfromkbyk=4(j−1)+a,j≥1,0≤a4,andB(j)aaregivenasfollows:212-213231@@@@R314@@@@R3124-234K00Figure1:Level1perfectcrystalB(Λ2)forA(1)3Ifjisodd,B(j)0=n34o,B(j)1=B(j)0∪n13o,B(j)2=B(j)1∪n14o,B(j)3=B(j)2∪n23,24o.Ifjiseven,B(j)0=n12o,B(j)1=B(j)0∪n13o,B(j)2=B(j)1∪n23o,B(j)3=B(j)2∪n14,24o.Thispaperisorganizedasfollows.Inthenextsection,wereviewperfectcrystals,Demazurecrystalsandthecriteriondevelopedin[KMOU].Fromsec-tion2to8,welistthesequenceofWeylgroupelements{w(k)},thesubsetB(j)a⊂BandotherimportantdataforDemazurecrystals.Inthelastsection,someobservationsandconjecturesaregiven.1CrystalsforDemazuremodulesIn[KMOU],acriterionforDemazurecrystalsispresented.Itclarifiestheirtensorproductstructure,andinvolvesaparameterκwhichmeasuresthedegreeofmixing.Herewesummarizethecriterionforκ=1.1.1PerfectcrystalWefollowthenotationsofthequantizeduniversalenvelopingalgebraandthecrystalbasein[KMN1],exceptthatweshalluseadifferentfontforthecrystalbase(L(λ),B(λ))oftheirreduciblehighestweightmoduleV(λ)inordertoavoidtheconfusionwiththecrystalbase(L,B)ofVinModf(g,Pcl)which3mayalsohaveanargument.Wereviewnecessarypropertiesofperfectcrystals.Ourmainreferenceis[KMN1].LetBbeaperfectcrystaloflevell.Thenforanyλ∈(P+cl)l,thereexistsauniqueelementb(λ)∈Bsuchthatϕ(b(λ))=λ.Letσbetheautomorphismof(P+cl)lgivenbyσλ=ε(b(λ)).Wesetbk=b(σk−1λ)andλk=σkλ.Thenperfectnessassuresthefollowingisomorphismofcrystals.B(λk−1)≃B(λk)⊗B.(1.1)DefinethesetofpathsP(λ,B)byP(λ,B)={p=···⊗p(2)⊗p(1)|p(j)∈B,p(k)=bkfork≫1}.Iteratingtheisomorphism(1.1),weseeB(λ)isisomorphictoP(λ,B).Underthisisomorphism,thehighestweightvectoruλinB(λ)correspondstothepathp=···⊗bk⊗···⊗b2⊗b1,whichwecallthegroundstatepath.Theactionsof˜eiand˜fionP(λ,B)aredeterminedbysignaturerule,whichweexplaininthenextsubsection.1.2SignatureruleWeneedtoknowtheactio