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arXiv:math/0503149v3[math.CO]5Mar2008S´eminaireLotharingiendeCombinatoire52(2004),Article-SOMECONJECTURESFORMACDONALDPOLYNOMIALSOFTYPEB,C,DMichelLassalleCentreNationaldelaRechercheScientifiqueInstitutGaspardMonge,Universit´edeMarne-la-Vall´ee77454Marne-la-Vall´eeCedex,FranceDedicatedtoAlainLascouxontheoccasionofhis60thbirthdayAbstract.WepresentconjecturesgivingformulasfortheMacdonaldpolynomialsoftypeB,C,Dwhichareindexedbyamultipleofthefirstfundamentalweight.Thetransitionmatricesbetweentwodifferenttypesareexplicitlygiven.IntroductionAmongsymmetricfunctions,thespecialimportanceofSchurfunctionscomesfromtheirintimateconnectionwithrepresentationtheory.ActuallytheirreduciblepolynomialrepresentationsofGLn(C)areindexedbypartitionsλ=(λ1,...,λn)oflength≤n,andtheircharactersaretheSchurfunctionssλ.Intheeighties,I.G.MacdonaldintroducedanewfamilyofsymmetricfunctionsPλ(q,t).Theseorthogonalpolynomialsdependrationallyontwoparametersq,tandgeneralizeSchurfunctions,whichareobtainedfort=q[9,10].Whentheindexingpartitionisreducedtoarow(k)(i.e.haslengthone),theMacdonaldpolynomialgk(q,t)ofnvariablesx=(x1,...,xn)aregivenbytheirgeneratingfunctionnYi=1(tuxi;q)∞(uxi;q)∞=Xr≥0urgr(x;q,t),withthestandardnotation(a;q)∞=Q∞i=0(1−aqi).Ofcoursefort=qthecompletefunctionss(r)=hrarerecovered.Afewyearslater,generalizinghispreviouswork,Macdonaldintroducedanotherclassoforthogonalpolynomials,whichareLaurentpolynomialsinseveralvariables,andgeneralizetheWeylcharactersofcompactsimpleLiegroups[11,12].Inthemostsimplesituationofthisnewframework,afamilyP(R)λ(q,t)ofpolynomials,dependingrationallyontwoparametersq,t,isattachedtoeachrootsystemR.TypesetbyAMS-TEX12MICHELLASSALLETheseorthogonalpolynomialsareelementsofthegroupalgebraoftheweightlatticeofR,invariantundertheactionoftheWeylgroup.TheyareindexedbythedominantweightsofR.WhenRisoftypeA,theorthogonalpolynomialsP(R)λ(q,t)correspondtothesymmetricfunctionsPλ(q,t)previouslystudiedin[9,10].Fort=q,theycorrespondtotheWeylcharactersχ(R)λofcompactsimpleLiegroups.ThispaperisonlydevotedtotheMacdonaldpolynomialswhichareindexedbyamultipleofthefirstfundamentalweightω1.SinceH.Weyl[15],itiswellknownthatχ(R)rω1isgivenby(i)hr(X)+hr−1(X),whenR=Bn,(ii)hr(X),whenR=Cn,(iii)hr(X)−hr−2(X),whenR=Dn,withX=(x1,...,xn,1/x1,...,1/xn).However,asfarastheauthorisaware,nosuchresultisknownwhent6=q,andnoexplicitexpansionisavailablefortheMacdonaldpolynomialsP(R)rω1(q,t).Thepurposeofthispaperistopresentsomeconjecturesgeneralizingthepreviousformulas.Actuallythisproblemcanbeconsideredinamoregeneralsetting,allowingtwodistinctparameterst,T,eachofwhichisattachedtoalengthofroots.WegiveanexplicitformulaforP(R)rω1(q,t,T)whenRisoftypeB,C,D,togetherwithanexplicitformulaforthetransitionmatricesbetweendifferenttypes.Theentriesofthesetransitionmatricesappeartobefullyfactorizedandrevealdeepconnectionswithbasichypergeometricseries.Weprovidesomesupportfortheseconjecturesbyshowingthattheyareverifieduponprincipalspecialization.Ontheotherhand,computercalculationsshowaverystrongempiricalevidenceintheirfavor.1.MacdonaldpolynomialsInthissectionweintroduceournotations,andrecallsomegeneralfactsaboutMacdonaldpolynomials.Formoredetailsthereaderisreferredto[11,12,13].ThemostgeneralclassofMacdonaldpolynomialsisassociatedwithapairofrootsystems(R,S),spanningthesamevectorspaceandhavingthesameWeylgroup,withRreduced.Hereweshallonlyconsiderthecaseofapair(R,R),withRoftypeB,C,D.LetVbeafinite-dimensionalrealvectorspaceendowedwithapositivedefinitesymmetricbilinearformhu,vi.Forallv∈V,wewrite|v|=hv,vi1/2,andv∨=2v/|v|2.MACDONALDPOLYNOMIALS3LetR⊂Vbeareducedirreduciblerootsystem,WtheWeylgroupofR,R+thesetofpositiveroots,{α1,...,αn}thebasisofsimpleroots,andR∨={α∨|α∈R}thedualrootsystem.LetQ=Pni=1ZαiandQ+=Pni=1NαibetherootlatticeofRanditspositiveoctant.LetP={λ∈V|hλ,α∨i∈Z∀α∈R}andP+={λ∈V|hλ,α∨i∈N∀α∈R+}betheweightlatticeofRandtheconeofdominantweights.AbasisofQisformedbythesimplerootsαi.AbasisofPisformedbythefundamentalweightsωidefinedbyhωi,α∨ji=δij.ApartialorderisdefinedonPbyλ≥μifandonlyλ−μ∈Q+.LetAdenotethegroupalgebraoverRofthefreeAbeliangroupP.Foreachλ∈PleteλdenotethecorrespondingelementofA,subjecttothemultiplicationruleeλeμ=eλ+μ.Theset{eλ,λ∈P}formsanR-basisofA.TheWeylgroupWactsonPandonA.LetAWdenotethesubspaceofW-invariantsinA.Suchelementsarecalled“symmetricpolynomials”.TherearetwoimportantexamplesofabasisofAW.Thefirstoneisgivenbytheorbit-sumsmλ=Xμ∈Wλeμ,λ∈P+.AnotherbasisisprovidedbytheWeylcharactersdefinedasfollows.Letδ=Yα∈R+(eα/2−e−α/2)=e−ρYα∈R+(eα−1),withρ=12Pα∈R+α∈P.Thenwδ=ε(w)δforanyw∈W,whereε(w)=det(w)=±1.Forallλ∈P,χλ=δ−1Xw∈Wε(w)ew(λ+ρ)isinAW,andtheset{χλ,λ∈P+}formsanR-basisofAW.Let0q1.Foranyindeterminatexandforallk∈N,define(x;q)∞=∞Yi=0(1−xqi),(x;q)k=k−1Yi=0(1−xqi).Foreachα∈Rlettα=qkαbeapositiverealnumbersuchthattα=tβif|α|=|β|.Thenwehaveatmosttwodifferentvaluesforthetα’s.Defineρk=12Xα∈R+kαα,ρ∗k=12Xα∈R+kαα∨.4MICHELLASSALLEIff=Pλ∈Paλeλ∈A,letf=Pλ∈Paλe−λand[f]1itsconstantterma0.TheinnerproductdefinedonAbyhf,giq,t=1|W|[f¯gΔq,t]1,with|W|theorderofW,andΔq,t=Yα∈R(eα;q)∞(tαeα;q)∞isnondegenerateandW-invariant.