Geometric positivity in the cohomology of homogene

arXiv:math/0610538v1[math.AG]18Oct2006GEOMETRICPOSITIVITYINTHECOHOMOLOGYOFHOMOGENEOUSSPACESANDGENERALIZEDSCHUBERTCALCULUSIZZETCOSKUNANDRAVIVAKILAbstract.Wedescriberecentworkonpositivedescriptionsofthestructureconstantsofthecoho-mologyofhomogeneousspacessuchastheGrassmannian,bydegenerationsandrelatedmethods.Wegivevariousextensionsoftheserules,somenewandconjectural,toK-theory,equivariantcohomology,equivariantK-theory,andquantumcohomology.Contents1.Introduction21.1.(TypeA)Grassmanniansandpositivity31.2.Advantagesofgeometricrules41.3.Flagvarieties61.4.Othergroups6Part1.TYPEARULES,USINGASPECIFICDEGENERATIONORDER82.TheGrassmannian82.1.Akeyexample82.2.Thedegenerationsingeneral112.3.ThegeometricLittlewood-Richardsonruleintheguiseofpartiallycompletedpuzzles112.4.Connectiontotableaux133.TheK-theory(orGrothendieckring)oftheGrassmannian144.TheequivariantcohomologyoftheGrassmannian154.1.Aworkedequivariantexample175.TheequivariantK-theoryoftheGrassmannian185.1.ExtendingExample4.1toequivariantK-theory196.AconjecturalgeometricLittlewood-Richardsonruleforthetwo-stepflagvariety207.Buch’sconjecturalcombinatorial(non-geometric)rulesinthethree-stepcase,andforthetwo-stepcaseinequivariant8.AlessexplicitconjecturalgeometricLittlewood-Richardsonruleforpartialflagvarietiesingeneral22Part2.MOREGENERALRULES,MOREGENERALDEGENERATIONORDERS239.Thecohomologyofflagvarieties239.1.PaintedMondriantableaux2910.QuantumcohomologyofGrassmanniansandflagvarieties3211.Linearspacesandaquadraticform36References38Date:Tuesday,October17,2006.2000MathematicsSubjectClassification.Primary:14M15,14N15;Secondary:14N10,14C17,14P99,05E10,05E05.DuringthepreparationofthisarticlethesecondauthorwaspartiallysupportedbyNSFGrantDMS-0238532.12COSKUNANDVAKIL1.IntroductionInthisarticlewedescriberecentworkonpositivealgorithmsforcomputingthestructureconstantsofthecohomologyofhomogeneousvarieties.WealsodiscussextensionsoftheserulestoK-theory,equivariantcohomology,equivariantK-theoryandquantumcohomology.Wehavetwoaims.First,wewouldliketoprovidetheexpertsinthefieldwithacompendiumofrecentresultsandreferencesregardingpositivityinSchubertcalculus.Second,wewouldliketopresentmanyexamplessothatthecasualusercanperformbasiccalculationsthatoccurinconcreteproblems.Homogeneousvarietiesareubiquitousinmathematics,playinganimportantroleinrepresentationthe-ory,algebraicanddifferentialgeometry,combinatoricsandthetheoryofsymmetricfunctions.Thestruc-tureconstants(“Littlewood-Richardsoncoefficients”)ofthecohomologyringsofhomogeneousvarietiesexhibitarichandsurprisingstructure.Forfundamentalgeometricreasons,theLittlewood-Richardsoncoefficientsandtheirgeneralizationstendtobepositive(interpretedappropriately).Incohomology,thisisaconsequenceofKleiman’sTransversalityTheorem1.3(sometimescalledtheKleiman-BertiniTheo-rem).InK-theory,themostnotablegeneralpositivityresultisatheoremofBrion[Br],andinequivariantcohomology,thekeyresultisduetoGraham[G],confirmingaconjectureofD.Peterson.Suchpositivitysuggeststhatthesecoefficientshaveacombinatorialinterpretation(a“Littlewood-Richardsonrule”),andthatsuchaninterpretationshouldbegeometricinnature.Inrecentyearspositivealgorithmsforcomputingtheseconstantshaveunraveledsomeofthisbeautifulstructure.WenowsurveythetechniquesusedinobtainingpositivegeometricalgorithmsfordeterminingthesestructureconstantsstartingwiththecaseoftheordinaryGrassmannians.InPart1weconsiderthecaseoftypeA,followingaparticularseriesofdegenerations(whichfirstarosein[V2],andaredescribedin§2.2)thatseemstobeparticularlyfruitfulinresolvingaproductofSchubertclassesintoacombinationofotherSchubertclasses.Inmostcases,wemayatleastconjecturallyinterpretthesedegenerationsintermsofgeneralizationsoftheelegantpuzzlesofA.KnutsonandT.Tao.ManyofthenewconjecturalstatementsinthisPartarejointworkofKnutsonandthesecondauthor.InPart2,insteadofdegeneratingaccordingtoafixedorder,weadaptourdegenerationordertotheSchubertproblemathand.Thisflexibilityallowsustosimplifythegeometry.InadditiontoanewruleforGrassmannians(§9),weobtainLittlewood-Richardsonrulesfortwo-stepflagvarieties(§9.1)andthequantumcohomologyofGrassmannians(§10).Furthermore,thesamedegenerationtechniquecanbeappliedtoFanovarietiesofquadrichypersurfaces,thusyieldingamethodtocalculatecertainintersectionsinTypeBandDGrassmannians(§11).Wewillpresentlotsofexplicitexamples,whicharetheheartofthearticle,astheyillustratethegeneraltechniques.Insomesense,everyexampleisageneralizationofasingleclassicalexample,§1.1.Weemphasizethatthisisanactive,burgeoningareaofresearch,andwearepresentingonlyasampleofrecentwork.Wewishtoatleastadvertiseseveralresultscloselyrelatedtothosediscussedhere.(i)L.MihalceahasanexplicitstatementofageometricLittlewood-RichardsonrulefortheLagrangianGrassmannian(typeC),verifiedcomputationallyinaconvincingnumberofcases.Hehasprovedpartofthestatementandisworkingontherest.(Thereisalreadyanon-geometricLittlewood-Richardsonruleinthiscase,duetoStembridge.)(ii)D.DavisispursuingongoingworkonananalogousgeometricruleintypeB.(iii)W.GrahamandS.Kumarhaveexplicitlycomputedthestructureco-efficientsof