Excited Young diagrams and equivariant Schubert ca

arXiv:math/0703637v1[math.AG]21Mar2007EXCITEDYOUNGDIAGRAMSANDEQUIVARIANTSCHUBERTCALCULUSTAKESHIIKEDAANDHIROSHINARUSEAbstract.Wedescribethetorus-equivariantcohomologyringofisotropicGrassman-niansbyusingalocalizationmaptothetorusfixedpoints.WepresenttwotypesofformulasforequivariantSchubertclassesofthesehomogeneousspaces.Thefirstformulainvolvescombinatorialobjectswhichwecall“excitedYoungdiagrams”andthesecondoneiswrittenintermsoffactorialSchurQ-orP-functions.Asanapplication,wegiveaGiambelli-typeformulafortheequivariantSchubertclasses.WealsogivecombinatorialandPfaffianformulasforthemultiplicityofasingularpointinaSchubertvariety.1.IntroductionInthispaper,wegiveexplicitdescriptionsoftheSchubertclassesinthe(torus)equi-variantcohomologyringoftheGrassmanniansaswellasthemaximalisotropicGrass-manniansofbothsymplecticandorthogonaltypes.OurmainresultsexpresstheimageofanequivariantSchubertclassunderthelocalizationmaptothetorusfixedpoints.Nowletusfixsomenotation.LetGbeacomplexsemisimpleconnectedalgebraicgroup.ChooseamaximaltorusTofGandaBorelsubgroupBcontainingT.LetPbeamaximalparabolicsubgroupofGcontainingB.Weareinterestedinthe(integral)T-equivariantcohomologyringH∗T(G/P)ofthehomogeneousspaceG/P.TheequivariantSchubertclassesareparametrizedbythesetWPofminimallengthrepresentativesforW/WP,whereWistheWeylgroupsofGandWPistheparabolicsubgroupassociatedtoP.ThesetWPalsoparametrizestheT-fixedpoints(G/P)TinG/P.Infactifweputev=vP(v∈WP)then(G/P)P={ev}v∈WP.LetB−denotetheoppositeBorelsubgroupsuchthatB−∩B=T.DefinetheSchubertvarietyXwassociatedwiththeelementw∈WP,tobetheclosureofB−-orbitB−ewofew.NotethatthecodimensionofXwinG/Pisℓ(w),thelengthofw,andev∈Xwifandonlyifw≤v,where≤isthepartialorderonWPinducedbytheBruhat-ChevalleyorderingofW.SinceXwisaT-stablesubvarietyinG/P,itinducesaT-equivariantfundamentalclass,theequivariantSchubertclass,denotedby[Xw]∈H2ℓ(w)T(G/P).Ourmaingoalistodescribe[Xw]explicitly.Inthispaper,weconsiderG/Pinthefollowinglist:•TypeAn−1:SL(n)/Pd(1≤d≤n),•TypeBn:SO(2n+1)/Pn,•TypeCn:Sp(2n)/Pn,•TypeDn:SO(2n)/Pd(d=n−1,n)wherewedenotebyPdthemaximalparabolicsubgroupassociatedtothed-thsimpleroot(thesimplerootsbeingindexedasin[5]).ItiswellknownthatthespaceSL(n)/PdcanbeidentifiedwiththeGrassmannianGd,nofd-dimensionalsubspacesinCn.Anyother12TAKESHIIKEDAANDHIROSHINARUSEG/PintheabovelistisamaximalisotropicGrassmannianwithrespecttoanorthogonalorsymplecticform(seeSection3fordetails).Ourdescriptionisbasedontheringhomomorphismι∗:H∗T(G/P)−→H∗T((G/P)T)=Mv∈WPH∗T(ev),inducedbytheinclusionι:(G/P)T֒→G/P.Thisι∗isknowntobeinjectiveandcalledthelocalizationmap.EachsummandH∗T(ev)iscanonicallyisomorphictothesymmetricalgebraS=SymZ(ˆT)ofthecharactergroupˆTofthetorusT.ThustheequivariantSchubertclass[Xw]isdescribedbyalist{[Xw]|v}v∈WPofpolynomialsinS,where[Xw]|vdenotetheimageoftheequivariantSchubertclass[Xw]underthehomomorphismι∗v:H∗T(G/P)−→H∗T(ev)inducedbytheinclusionιv:{ev}֒→G/P.IntypeAn−1case(cf.Theorem2),KnutsonandTao[12]discoveredthat[Xw]|vcanbeidentifiedwithasuitablyspecialized‘factorial’Schurfunction,amulti-parameterdeformationofSchurfunction(seeSection5forthedefinition).TheirargumentusesaremarkablevanishingpropertyofthefactorialSchurfunction(cf.Proposition5).Byatotallydifferentmethod,Lakshmibai,Raghavan,andSankaran[17]showedthesameresult,althoughtheydonotstateitexplicitlyintermsoffactorialSchurfunction.Infact,theystartfromacombinatorialexpressionfor[Xw]|vintermsofasetofnon-intersectingpaths,whichcomesfromadetailedanalysisofGr¨obnerbasisofthedefiningidealoftheSchubertvarietyduetoKreimanandLakshmibai[15],andKodiyalamandRaghavan[10],andthenrewritetheexpressionintoaratioofsomedeterminants,whichisaformoffactorialSchurfunction.TypeCn,thecaseofLagrangianGrassmannian,wasstudiedinapaper[8]bythefirstnamedauthor,where[Xw]|visexpressedintermsoffactorialSchurQ-functiondefinedbyIvanov[9].TheproofisacomparisonofPieri-Chevalleytyperecurrencerelationsforboth[Xw]|vandthefactorialSchurQ-function.ThisstrategyofidentificationgoeswellforotherG/Pinourlistabove.Actually,weproveinthispapertheanalogousresultfortypesBnandDn,theorthogonalGrassmannian,i.e.,wepresentaformulafor[Xw]|vintermsoffactorialSchurP-functionforthesespaces(Theorem4).Asanapplicationoftheseformulas,weobtainedanGiambelli-typeformula(Corollary2)forisotropicGrassmanniansthatexpressesanarbitraryequivariantSchubertclassasaPfaffianofSchubertclassesassociatedwiththe‘two-row’(strict)partitions.ThisformulaisanequivariantanalogueoftheGiambelliformuladuetoPragacz[20]inthecaseofordinarycohomology.Anothertypeofformulas(Theorem1,Theorem3)wediscussinthepresentpaperinvolvescombinatorialobjects,whichwecallexcitedYoungdiagrams,EYDsforshort.TheideaofEYDswasinspiredbythework[17]ofLakshmibai,Raghavan,andSankaranmentionedabove.Theseformulashavea‘positive’natureinthesensethatitisexpressedasasumoverasetofEYDswitheachsummandbeingaproductsofsomepositiveroots.Foranexposition,hereweconsiderGd,n.Itiswell-knownthatthesetWPdisparametrizedbythesetofpartitionλ=(λ1,...,λd)suchthatn−d≥λ1≥···≥λd≥0,orequiva-lentlytheYoungdiagramscontainedintherectangularofshaped×(