MULTIPLICITY ON A RICHARDSON VARIETY IN A COMINUSCULE G P

Description

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].

• positive roots


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• subgroup associated


• schubert variety


• borel subgroup


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• also been

Sujets

• Richardson
• Root system
• Schubert variety
• Borel subgroup

Informations

• Publié par: pefav
• Nombre de lectures: 24
• Langue: English

Extrait

MULTIPLICITYONARICHARDSONVARIETYINA COMINUSCULEG/P MICHAE ¨ LBALAN Abstract.WeshowthatinacominusculepartialflagvarietyG/P,themul- tiplicityofanarbitrarypointonaRichardsonvarietyX w v =X w ∩ X v ⊂ G/P istheproductofitsmultiplicitiesontheSchubertvarietiesX w andX v . Introduction Richardsonvarieties,namedafter[33],areintersectionsofaSchubertvariety andanoppositeSchubertvarietyinsideapartialflagvarietyG/P(Gaconnected complexsemi-simplegroup,Paparabolicsubgroup).Theypreviouslyappearedin [9,Ch.XIV,§4]and[36],aswellasthecorrespondingopencellsin[6].Theyhave sinceplayedaroleindifferentcontexts,suchasequivariantK-theory[24],positivity inGrothendieckgroups[3],standardmonomialtheory[4],Poissongeometry[8], positroidvarieties[13],andtheirgeneralizations[14,1]. Ontheotherhand,singularitiesofSchubertvarietieshavebeenextensivelystud- iedinthelastdecades.ThesingularlocusofSchubertvarietiesinGrassmannians hasbeendeterminedindependentlyin[37]and[27],andmoregenerallyinami- nusculeG/Pin[26].InthefullflagvarietyoftypeA n ,ithasbeendetermined independentlyin[2],[5],[12],and[29]. Moreover,themultiplicityofasingularpointonaSchubertvarietyisknownin severalcases:whenG/Pisminusculeofarbitrarytype,orcominusculeoftypeC n , arecursiveformulawasgivenin[26].Adirectdeterminantalformulawasgivenin [34]forG/PaGrassmannian;ithasbeensubsequentlyinterpretedintermsofnon- intersectinglatticepaths[17].Themultiplicityproblemhasalsobeenstudiedin relationshipwithHilbertfunctionsandGro¨bnerdegenerations[7,16,18,23,31,32], aswellaswithT–equivariantcohomology[10,11,15,20,21,25].Theproblemof determiningthemultiplicityofapointinaSchubertvarietyinthefullflagvariety ismorecomplicated;see[39,28,40,41]. ForRichardsonvarietiesinaminusculeG/P,themultiplicityofaT-fixedpoint (T⊂PamaximaltorusinG)hasbeendeterminedbyKreimanandLakshmibai [22](fortheGro¨bnerpointofview,seealso[19]intypeA n and[38]inorthogonal types). Inthispaper,wedeterminethemultiplicityofanarbitrarypoint 1 onaRichard- sonvarietyinacominusculeG/P. Beforestatingthemainresult,letusfixsomenotation.LetG,P,Tbeasabove, withGadjoint.LetX(T)bethecharactergroupofT,R ⊂ X(T)therootsystem, Date:April21,2011. 2010MathematicsSubjectClassification.Primary14M15;Secondary14B0514L30. 1 NotethatunlikeinthecaseofaSchubertvariety,thisnolongerfollowsfrominformation aboutT-fixedpoints,aspointedoutintheintroductionsof[22]and[19]. 1