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arXiv:0710.0268v4[math.RT]2Mar2008SCHURTYPEFUNCTIONSASSOCIATEDWITHPOLYNOMIALSEQUENCESOFBINOMIALTYPEMINORUITOHAbstract.WeintroduceaclassofSchurtypefunctionsassociatedwithpolynomialsequencesofbinomialtype.ThiscanberegardedasageneralizationoftheordinarySchurfunctionsandthefactorialSchurfunctions.Thisgeneralizationsatisfiessomeinterestingexpansionformulas,inwhichthereisacuriousduality.MoreoverthisclassincludesexampleswhichareusefultodescribetheeigenvaluesofCapellitypecentralelementsoftheuniversalenvelopingalgebrasofclassicalLiealgebras.IntroductionInthisarticle,weintroduceaclassofSchurtypefunctionsassociatedwithpolynomialsequencesofbinomialtype.Namely,suggestedbythedefinitionoftheordinarySchurfunctiondet(xλi+N−ij)/det(xN−ij),weconsiderthefollowingSchurtypefunction:det(pλi+N−i(xj))/det(pN−i(xj)).Here{pn(x)}n≥0isapolynomialsequenceofbinomialtype.ThiscanberegardedasageneralizationoftheordinarySchurfunctionsandmoreoverthefactorialSchurfunctions([BL],[CL]).Inadditiontothis,wealsoconsiderthefollowingfunction:det(p∗λi+N−i(xj))/det(p∗N−i(xj)).Hereweputp∗n(x)=x−1pn+1(x)(thisp∗n(x)isapolynomial,andsatisfiesgoodrelations;seeSection1.3).Themainresultsofthisarticlearesomeexpansionformulasforthesefunctionsandtheirmysteriousdualitycorrespondingtotheexchangepn(x)↔p∗n(x)andtheconjugationofpartitions(Sections3,4,5,and6).Mostofthemareprovedbyelementaryandstraightforwardcalculations.Besidetheseresults,wealsogiveanapplicationtorepresentationtheoryofLiealgebras(Section8).Letusbrieflyexplainthisapplication.ThefactorialSchurfunctionsareusefultoex-presstheeigenvaluesofCapellitypecentralelementsoftheuniversalenvelopingalgebrasofthegenerallinearLiealgebra(moreprecisely,weshouldsaythatthe“shiftedSchurfunctions”areuseful;bytheshiftofvariables,thefactorialSchurfunctionsaretrans-formedintotheshiftedSchurfunctions([OO1],[O])).Inthisarticle,weaimtointroducesimilarSchurtypefunctionswhichisusefultoexpresstheeigenvaluesofCapellitypecentralelementsoftheuniversalenvelopingalgebrasoftheorthogonalandsymplecticLiealgebras.Thisaimisachievedinthecaseofthepolynomialsequencecorrespondingtothecentraldifference.TheseSchurtypefunctionsarealsorelatedwiththeanaloguesoftheshiftedSchurfunctionsgivenin[OO2],whichwereintroducedwithasimilaraim.Ourclassisanothernaturalgeneralizationcontainingtheseremarkableexamples.2000MathematicsSubjectClassification.Primary05E05,05A40;Secondary17B35,15A15;Keywordsandphrases.Schurfunctions,polynomialsequenceofbinomialtype,centralelementsofuniversalenvelopingalgebras.12MINORUITOHVariousgeneralizationsareknownfortheSchurfunction.Manyofthemareobtainedbyreplacingtheordinarypowersbysomepolynomialsequence(furthergeneralizationsareknown;see[M2]).Inparticular,thegeneralizationassociatedwiththepolynomialsintheformpn(x)=Qnk=1(x−ak)iswellknown[M1],andthiscontainsthefactorialSchurfunction.Inthisarticle,weconsideranothergeneralizationwhichisnotparticularlylargebutincludesinterestingphenomenaandexamples.1.polynomialsequencesofbinomialtypeFirst,werecallthepropertiesofpolynomialsequencesofbinomialtype.See[MR],[R],[RKO],and[S]forfurtherdetails.1.1.Westartwiththedefinition.Apolynomialsequence{pn(x)}n≥0inwhichthedegreeofeachpolynomialisequaltoitsindex,issaidtobeofbinomialtype,whenthefollowingrelationholdsforanyn≥0:pn(x+y)=Xk≥0nkpk(x)pn−k(y).Letusseesomeexamples.First,thesequence{xn}n≥0oftheordinarypowersisofbinomialtype,becausewehavetherelation(x+y)n=Xk≥0nkxkyn−k(theordinarybinomialexpansion).Asothertypicalexamples,somefactorialpowersarewellknown.Wedefinetherisingfactorialpowerxnandfallingfactorialpowerxnbyxn=x(x+1)···(x+n−1),xn=x(x−1)···(x−n+1).Then{xn}n≥0and{xn}n≥0arealsoofbinomialtype.Indeedthefollowingrelationshold([MR],[RKO]):(x+y)n=Xk≥0nkxkyn−k,(x+y)n=Xk≥0nkxkyn−k.Itiseasilyseenthatpn(0)=δn,0,when{pn(x)}n≥0isofbinomialtype.1.2.Anaturalcorrespondenceisknownbetweenpolynomialsequencesofbinomialtypeanddeltaoperators[RKO].Letusrecallthedefinitionofdeltaoperators.AlinearoperatorQ=Qx:C[x]→C[x]iscalleda“deltaoperator,”whenthefollowingtwopropertieshold:(i)Qreducesdegreesofpolynomialsbyone;(ii)Qisshift-invariant(namely,QcommuteswithallshiftoperatorsEa:f(x)7→f(x+a)).AtypicalexampleisthedifferentiationD=ddx.MoreovertheforwarddifferenceΔ+andthebackwarddifferenceΔ−aredeltaoperators:Δ+:f(x)7→f(x+1)−f(x),Δ−:f(x)7→f(x)−f(x−1).EverydeltaoperatorcanbewrittenasapowerseriesofthedifferentiationDinthefollowingformwitha1,a2,...∈C,a16=0:Q=a1D+a2D2+a3D3+···.SCHURFUNCTIONSFORPOLYNOMIALSEQUENCESOFBINOMIALTYPE3Thereisanaturalone-to-onecorrespondencebetweenthesedeltaoperatorsandpoly-nomialsequencesofbinomialtype.Thesearerelatedviatherelation(1.1)Qpn(x)=npn−1(x).Namely,forapolynomialsequenceofbinomialtype{pn(x)}n≥0,thelinearoperatorQ:C[x]→C[x]determinedby(1.1)isadeltaoperator.Conversely,foranydeltaopera-torQ,apolynomialsequence{pn(x)}n≥0isuniquelydeterminedby(1.1)andtherelationpn(0)=δn,0,andthis{pn(x)}n≥0isofbinomialtype(thesearecalledbasicpolynomials).Forexample,thedifferentiationD=ddxcorrespondstothesequence{xn}n≥0,becauseDxn=nxn−1.Similarly,theforwarddifferenceΔ+andthebackwarddi