A quantum type deformation of the cohomology ring

arXiv:math/0610298v3[math.CO]2Jan2007AQUANTUMTYPEDEFORMATIONOFTHECOHOMOLOGYRINGOFFLAGMANIFOLDSAUGUSTIN-LIVIUMAREAbstract.Letq1,...,qnbesomevariablesandsetK:=Z[q1,...,qn]/hq1...qni.WeshowthatthereexistsaK-bilinearproduct⋆onH∗(Fn,Z)⊗Kwhichisuniquelydeterminedbysomequantumcohomologylikeproperties(mostimportantly,adegreetworelationinvolvingthegeneratorsandananalogueoftheflatnessoftheDubrovinconnection).Thenweprovethat⋆satisfiestheFrobeniuspropertywithrespecttothePoincar´epairingofH∗(Fn,Z);thisleadsimmediatelytotheorthogonalityofthecorrespondingSchuberttypepolynomials.Wealsonotethatifwepickk∈{1,...,n}andweformallyreplaceqkby0,thering(H∗(Fn,Z)⊗K,⋆)becomesisomorphictotheusualsmallquantumcohomologyringofFn,byanisomorphismwhichisdescribedprecisely.1.IntroductionWeconsiderthecomplexflagmanifoldFn={V1⊂V2⊂...⊂Vn−1⊂Cn|VkisalinearsubspaceofCn,dimVk=k,k=1,...,n−1}.Foreveryk∈{1,...,n−1}weconsiderthetautologicalvectorbundleVkoverFnandthecohomology1classyk=−c1(detVk)∈H2(Fn).Itisknownthattheclassesy1,...,yn−1generatetheringH∗(Fn).Moreover,letussetxk:=yk−yk−1,1≤k≤n(whereweassigny0=yn:=0).IfY1,...,Yn−1aresomevariablesandwesetXk:=Yk−Yk−1likebefore,thenwehavetheBorelisomorphism(1)H∗(Fn)≃Z[Y1,...,Yn−1]/hnonconstantsymmetricpolynomialsinX1,...,Xni,whichmapsyitothecosetofYi,foralli∈{1,...,n−1}.Letusalsoconsiderthevariablesq1,...,qnandsetK:=Z[q1,...,qn]/hq1...qni.Themainresultofthispaperisasfollows.Theorem1.1.Thereexistsaproduct⋆onH∗(Fn)⊗Kwhichisuniquelydeterminedbythefollowingproperties:(i)⋆isK-bilinearDate:February2,2008.1ThecoefficientringforcohomologywillbealwaysZ,unlessotherwisespecified.12A.-L.MARE(ii)⋆preservesthegradinginducedbytheusualgradingofH∗(Fn)combinedwithdegqj=4,1≤j≤n(iii)⋆isadeformationoftheusualproduct,inthesensethatifweformallyreplaceallqjby0,weobtaintheusual(cup-)productonH∗(Fn)(iv)⋆iscommutative(v)⋆isassociative(vi)wehavetherelationX1≤ij≤nxi⋆xj+nXj=1qj=0(vii)thecoefficients(yi⋆a)d∈H∗(Fn)ofqd:=qd11...qdnninyi⋆asatisfy(di−dn)(yj⋆a)d=(dj−dn)(yi⋆a)d,foranyd=(d1,...,dn)≥0,a∈H∗(Fn),and1≤i,j≤n−1.Wewillalsoprovesomepropertiesoftheproduct⋆.ThefirstoneinvolvesthePoincar´epairing(,)onH∗(Fn);weactuallyextendittoaK-bilinearformonH∗(Fn)⊗K.Theorem1.2.Theproduct⋆satisfiesthefollowingFrobeniustypeproperty:(a⋆b,c)=(a,b⋆c)foranya,b,c∈H∗(Fn).Insection3wewillalsogiveapresentationofthering(H∗(Fn)⊗K,⋆)intermsofgeneratorsandrelations.Theorem1.2willallowustoprovethatrepresentativesofSchubertclassesinthisquotientringsatisfyacertainorthogonalityrelation,similartotheonesatisfiedbythequantumSchubertpolynomialsof[Fo-Ge-Po]and[Ki-Ma].Finallywewillprovethefollowingtheorem.Theorem1.3.Foranyfixedi∈{1,...,n},onedenotesbyhqiitheidealofH∗(Fn)⊗Kgeneratedbyqi.Thenthequotientring(H∗(Fn)⊗K,⋆)/hqiiisisomorphictotheactual(small)quantumcohomologyringofFn.Theisomorphismisrealizedintermsoftheformalreplacements˜y1:=yi−yi−1,...,˜yi−1:=yi−y1,˜yi:=yi,˜yi+1:=yi−yn−1,...,˜yn−1:=yi−yi+1and˜q1:=qi−1,˜q2:=qi−2,...,˜qi−1:=q1,˜qi:=qn,˜qi+1:=qn−1,...,˜qn−1:=qi+1,ifi6=n;ifi=n,thereplacementsare˜yj:=yj,˜qj:=qj,forall1≤j≤n−1.Moreprecisely,theisomorphismisthemapfromH∗(Fn)⊗K/hqii=Z[y1,...,yn−1,q1,...,qi−1,qi+1,...,qn]/S(x1,...,xn)toH∗(Fn)⊗Z[˜q1,...,˜qn−1]=Z[˜y1,...,˜yn−1,˜q1,...,˜qn−1]/S(˜x1,...,˜xn)AQUANTUMDEFORMATIONOFTHECOHOMOLOGYOFFLAGMANIFOLDS3describedbythereplacementsfromabove.Herexjstandsforyj−yj−1,˜xjfor˜yj−˜yj−1(whereyn=y0=˜yn=˜y0:=0),S(x1,...,xn)fortheidealgeneratedbythenon-constantsymmetricpolynomialsinx1,...,xn,andsimilarlyforS(˜x1,...,˜xn).Remark.Theproduct⋆consideredabovecouldberelevantinthecontextofthe(small)quantumcohomologyoftheinfinitedimensionalflagmanifoldLSU(n)/T(cf.[Gu-Ot]),whereLSU(n)isthespaceoffreeloopsinSU(n)andT⊂SU(n)isamaximaltorus.Forinstance,Theorem1.3isinspiredbyapropertyofthequantumcohomologyringofLSU(n)/Twhichwasconjecturedin[Gu-Ot](seetheremark(b)followingCorollary4.2).Thereexistssofarnorigorousdefinitionofthelatterring.In[Mar1]weconsideredaproduct◦onH∗(LSU(n)/T)⊗R[q1,...,qn]whichsatisfiescertainnatural,quantumcohomologylike,properties,similarto(i)-(vii)fromabove(seeTheorem1.1ofthatpaper).Itisanopenquestionwhethersuchaproduct◦exists.Thepaperisorganizedasfollows:firstweproveTheorem1.3andtheuniquenesspartofTheorem1.1.Thenweprovetheexistencepartofthelattertheorem,whichusesapresentationofthering(H∗(Fn)⊗K,⋆)intermsofgeneratorsandrelations.FinallyweprovetheFrobeniuspropertyandtheorthogonalityoftheSchuberttypepolynomials.Suchpolynomialsaredeterminedexplicitlyinthecasen=3.Acknowledgements.IthankMartinGuestandTakashiOtofujifordiscussionsaboutthetopicsofthepaper.2.Uniquenessoftheproduct⋆Wewillshowthatthereexistsatmostoneproduct⋆withtheproperties(i)-(vii)inTheorem1.1.Themaininstrumentisthefollowingresult,whichisaparticularcaseofTheorem1.2of[Mar2].Theorem2.1.([Mar2])LetY1,...,Yn−1,Q1,...,Qn−1besomevariablesandsetXj:=Yj−Yj−1for1≤j≤n,whereY0=Yn:=0.DenotebyItheidealofZ[Y1,...,Yn−1]generatedbythenon-constantsymmetricpolynomialsinX1,...,Xn.Thereexistsatmostoneproduct◦on(Z[Y1,...,Yn−1]/I)⊗Z[Q1,...,Qn−1]whichiscommutative,associative,Z[{Qi}]-bilinear,isadeformationofthecanon