MULTIPLICITY ON A RICHARDSON VARIETY IN A COMINUSCULE G P

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MULTIPLICITY ON A RICHARDSON VARIETY IN A COMINUSCULE G/P MICHAEL BALAN Abstract. We show that in a cominuscule partial flag variety G/P , the mul- tiplicity of an arbitrary point on a Richardson variety Xvw = Xw ?X v ? G/P is the product of its multiplicities on the Schubert varieties Xw and Xv . Introduction Richardson varieties, named after [33], are intersections of a Schubert variety and an opposite Schubert variety inside a partial flag variety G/P (G a connected complex semi-simple group, P a parabolic subgroup). They previously appeared in [9, Ch. XIV, 4] and [36], as well as the corresponding open cells in [6]. They have since played a role in different contexts, such as equivariant K-theory [24], positivity in Grothendieck groups [3], standard monomial theory [4], Poisson geometry [8], positroid varieties [13], and their generalizations [14, 1]. On the other hand, singularities of Schubert varieties have been extensively stud- ied in the last decades. The singular locus of Schubert varieties in Grassmannians has been determined independently in [37] and [27], and more generally in a mi- nuscule G/P in [26]. In the full flag variety of type An, it has been determined independently in [2], [5], [12], and [29].

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MULTIPLICITYONARICHARDSONVARIETYINACOMINUSCULEG/PMICHAE¨LBALANAbstract.WeshowthatinacominusculepartialﬂagvarietyG/P,themul-tiplicityofanarbitrarypointonaRichardsonvarietyXwv=Xw∩Xv⊂G/PistheproductofitsmultiplicitiesontheSchubertvarietiesXwandXv.IntroductionRichardsonvarieties,namedafter[33],areintersectionsofaSchubertvarietyandanoppositeSchubertvarietyinsideapartialﬂagvarietyG/P(Gaconnectedcomplexsemi-simplegroup,Paparabolicsubgroup).Theypreviouslyappearedin[9,Ch.XIV,§4]and[36],aswellasthecorrespondingopencellsin[6].Theyhavesinceplayedaroleindiﬀerentcontexts,suchasequivariantK-theory[24],positivityinGrothendieckgroups[3],standardmonomialtheory[4],Poissongeometry[8],positroidvarieties[13],andtheirgeneralizations[14,1].Ontheotherhand,singularitiesofSchubertvarietieshavebeenextensivelystud-iedinthelastdecades.ThesingularlocusofSchubertvarietiesinGrassmannianshasbeendeterminedindependentlyin[37]and[27],andmoregenerallyinami-nusculeG/Pin[26].InthefullﬂagvarietyoftypeAn,ithasbeendeterminedindependentlyin[2],[5],[12],and[29].Moreover,themultiplicityofasingularpointonaSchubertvarietyisknowninseveralcases:whenG/Pisminusculeofarbitrarytype,orcominusculeoftypeCn,arecursiveformulawasgivenin[26].Adirectdeterminantalformulawasgivenin[34]forG/PaGrassmannian;ithasbeensubsequentlyinterpretedintermsofnon-intersectinglatticepaths[17].ThemultiplicityproblemhasalsobeenstudiedinrelationshipwithHilbertfunctionsandGro¨bnerdegenerations[7,16,18,23,31,32],aswellaswithT–equivariantcohomology[10,11,15,20,21,25].TheproblemofdeterminingthemultiplicityofapointinaSchubertvarietyinthefullﬂagvarietyismorecomplicated;see[39,28,40,41].ForRichardsonvarietiesinaminusculeG/P,themultiplicityofaT-ﬁxedpoint(T⊂PamaximaltorusinG)hasbeendeterminedbyKreimanandLakshmibai[22](fortheGro¨bnerpointofview,seealso[19]intypeAnand[38]inorthogonaltypes).Inthispaper,wedeterminethemultiplicityofanarbitrarypoint1onaRichard-sonvarietyinacominusculeG/P.Beforestatingthemainresult,letusﬁxsomenotation.LetG,P,Tbeasabove,withGadjoint.LetX(T)bethecharactergroupofT,R⊂X(T)therootsystem,Date:April21,2011.2010MathematicsSubjectClassiﬁcation.Primary14M15;Secondary14B0514L30.1NotethatunlikeinthecaseofaSchubertvariety,thisnolongerfollowsfrominformationaboutT-ﬁxedpoints,aspointedoutintheintroductionsof[22]and[19].1

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