A characterization of interpolation Macdonald poly

arXiv:q-alg/9712052v123Dec1997ONNEWTONINTERPOLATIONOFSYMMETRICFUNCTIONS.ACHARACTERIZATIONOFINTERPOLATIONMACDONALDPOLYNOMIALS.AndreiOkounkovContents1.Briefintroduction2.Generalinterpolationproblem3.Examplesofinterpolationpolynomials3.1Universalinterpolationpolynomials3.2Factorialmonomialsymmetricfunctions3.3FactorialSchurfunctions3.4InterpolationMacdonaldpolynomials4.Statementofthecharacterizationtheorem5.Reductionsoftheproof6.Proofofthetheoreminthetwovariablescase(n=1)7.Proofofthetheoremforn1A.Appendix.TableofperfectgridsB.Appendix.Anexampleoftheuniversalinterpolationpolyno-mial1.BriefintroductionThepurposeofthispaperistocharacterizeinterpolationMacdonaldpolynomialsinsideaverygeneralNewtoninterpolationschemeforsymmetricpolynomials.ThisgeneralNewtoninterpolationproblemisdiscussedinSection2;itdependsasonaparameteronamapΩ(Z≥0)#ofvariablesΩ−−−−→groundfieldk,whichwecallagridink.Ourpresentunderstandingofthisgeneralproblemcanbedescribedasfollows:mostofitiscoveredbyanunexploredandmysteriousoceanTheauthorisgratefultotheMSRIinBerkeleyforhospitalityandtotheNSFforfinancialsupportduringhisstayattheMSRI(grantDMS–9022140).TypesetbyAMS-TEX12ANDREIOKOUNKOVformedbygenericgridsΩ(seeSection3.1).Inthemidstofthisabyssthereare3piecesofdryland,namelythe3followingexactlysolvablecases:(1)factorialmonomialsymmetricfunctions,(2)factorialSchurfunctions,(3)interpolationMacdonaldpolynomials,whicharedescribedinSections3.2–3.4.Thefirstcaseisjustdull,thesecondoneisstillratherelementary;bothofthemareparameterizedbyanarbitrarysequenceofpairwisedistinctelementsofthegroundfieldk.ThesetwocontinentsarejoinedbyabeautifularchipelagoofinterpolationMacdonaldpolynomials.Moreprecisely,byinterpolationMacdonaldpolynomialswemeanthesocalledBC-typeinterpolationMacdonaldpolynomials,introducedandstudiedin[Ok4].AsparticularcasesanddegenerationsthesepolynomialsincludepolynomialsstudiedbyF.Knop,G.Ol-shanski,S.Sahi,andtheauthorinalongseriesofpapers,seeReferences.Thesepolynomialsdependson5parametersofwhichonly3arenon-trivialbecauseofanactionofa2-dimensionalgroupofaffinetransformations.SomeofthepropertiesofthesemostremarkablepolynomialsarediscussedinSection3.4.Itisnaturaltoaskifanysimpleabstractpropertycharacterizesthe3aboveexactlysolvablecasesofourgeneralinterpolationproblem.Assuchapropertyweproposetheextravanishingproperty(4.2)whichsaysthattheNewtoninterpolationpolynomialsshouldvanishnotonlyatthosepointswheretheyaresupposedtovanishbytheirdefinitionbutalsoatcertainextrapoints“forfree”.Moreprecisely,thepolynomiallabeledbyapartitionμvanishesatthepointlabeledbyapartitionλunlessμislessorequaltoλinthepartialorderofpartitionsbyinclusionμ⊂λ.Wecallallgridsthatenjoythispropertyperfect.Thisextravanishingpropertycanbecomparedtothefollowingwell-knownpropertyofordinaryMacdonaldpolynomials.AlthoughtheGram-Schmidtor-thogonalizationprocessrequiresachoiceofatotalorderonthepolynomialstobeorthogonalized,theMacdonaldorthogonalpolynomialsdonotactuallydependonthechoiceofatotalorderonthemonomialsymmetricfunctionsaslongthistotalorderiscompatiblewiththepartialdominantorderofpartitions.1Also,theextravanishingpropertycanbecomparedtoawellknownphenomenoninintegrablesystemswheremanyexactlysolvablesystemshave“extra”integralsofmotion,thatis,moreintegralsofmotionthanisrequiredbythedefinitionofintegrability[Kr,CV].Itisinterestingtonoticethatcertainintegrablemany-bodysystemstowhichinterpolationMacdonaldpolynomialsareverycloselyconnectedwereconjecturedtobecharacterizedbythis“extra”integrabilityproperty[CV];later,however,certainnewexampleswerefoundin[CFV].Oursituationis,ofcourse,muchsimpler.Ourmainresult(Section4)isthatthethreeabovecasesplusdegenerationsofthethirdoneexhaustthesetofallperfectgrids.Asacorollaryofthistheorem,weconcludethatnoothergridΩadmitsa1Itwouldbeprobablyinterestingtodescribeallinterpolationororthogonalsymmetricpoly-nomialswhichsatisfysucha“extratriangularity”condition.INTERPOLATIONMACDONALDPOLYNOMIALS3tableauxsumformulaoftheform(4.1)fortheNewtoninterpolationpolynomials,nordoesitadmitanintegralrepresentationofinterpolationpolynomialsanalogoustotheq-integralrepresentationforinterpolationMacdonaldpolynomialsobtainedin[Ok4].Thatis,anynewexactlysolvablecaseofsymmetricNewtoninterpolationhastobebasedonsomeentirelynewtypeofformulas.TheproofofthischaracterizationtheoremisgiveninSections5–7.Section5containssomegeneralstatements,whereasthetwoothersectionsaredevotedtotheconsiderationofthemanypossiblecases.Thereexistalsonon-symmetricMacdonaldinterpolationpolynomialswhichformalinearbasisinthealgebraofallpolynomials,see[Kn,S2].Itisplausiblethatthosepolynomialsmighthaveasimilarcharacterization.NotealsothatacertaincharacterizationofordinaryMacdonaldpolynomialsinsidesomegeneralclassoforthogonalpolynomialswasfoundbyS.Kerovin[K].2.GeneralinterpolationproblemWeconsiderNewtoninterpolationofsymmetricpolynomialsinn+1variables(n=0,1,...,∞)withcoefficientsinsomeinfinitefieldk.Firstassumeforsimplic-itythatn∞;thecasen=∞willbecoveredattheendofthissection.Anynaturalbasisinthespaceofsymmetricpolynomialsofdegree≤disindexed