ON THE QUIVER GRASSMANNIAN IN THE ACYCLIC CASE

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ar X iv :m at h/ 06 11 07 4v 2 [m ath .R T] 9 Fe b 2 00 7 ON THE QUIVER GRASSMANNIAN IN THE ACYCLIC CASE PHILIPPE CALDERO AND MARKUS REINEKE Abstract. Let A be the path algebra of a quiver Q with no oriented cycle. We study geometric properties of the Grassmannians of submodules of a given A-module M . In particular, we obtain some sufficient conditions for smoothness, polynomial cardinality and we give different approaches to Euler characteristics. Our main result is the positivity of Euler characteristics when M is an exceptional module. This solves a conjecture of Fomin and Zelevinsky for acyclic cluster algebras. 0. Introduction Let M be a finite dimensional space on a field k. The Grassmannian Gre(M,k) of M is the set of subspaces of dimension e. It is well known that Gre(M,k) is an algebraic variety with nice properties. For instance, the linear group GLe(M,k) acts transitively on Gre(M,k) with parabolic stabilizer, hence the variety Gre(M,k) is smooth and projective. Suppose now that M has a structure of A-module, where A is a finitely generated k- algebra. It is natural to define the Grassmannian Gre(M,A) of A-submodules of M of given dimension e.

• called


• geometric properties


• laurent polynomial


• polynomial pe


• projective indecomposable representation


• quiver grassmannians


• grassmannians

Sujets

• Geometry
• Laurent polynomial
• Grassmannian

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 52
• Langue: English

Extrait

ONTHEQUIVERGRASSMANNIANINTHEACYCLICCASE PHILIPPECALDEROANDMARKUSREINEKE Abstract. Let A bethepathalgebraofaquiver Q withnoorientedcycle.Westudy geometricpropertiesoftheGrassmanniansofsubmodulesofagiven A -module M .In particular,weobtainsomesufficientconditionsforsmoothness,polynomialcardinality andwegivedifferentapproachestoEulercharacteristics.Ourmainresultisthepositivity ofEulercharacteristicswhenMisanexceptionalmodule.Thissolvesaconjectureof FominandZelevinskyforacyclicclusteralgebras. 0. Introduction LetMbeafinitedimensionalspaceonafieldk.TheGrassmannian Gr e (M,k)ofM isthesetofsubspacesofdimensione.Itiswellknownthat Gr e (M,k)isanalgebraic varietywithniceproperties.Forinstance,thelineargroup GL e (M,k)actstransitivelyon Gr e (M,k)withparabolicstabilizer,hencethevariety Gr e (M,k)issmoothandprojective. SupposenowthatMhasastructureofA-module,whereAisafinitelygeneratedk- algebra.ItisnaturaltodefinetheGrassmannian Gr e (M,A)ofA-submodulesofMof givendimensione.ItisaclosedsubvarietyofGr e (M,k),henceitisaprojectivevariety. Butingeneralthisvarietyisnotsmooth. LetQbeafinitequiverwithnoorientedcycleandletA:=kQbethepathalgebraof Q.Then,Aisafinitedimensionalalgebrawithminimalidempotentse i ,i∈Q 0 ,where Q 0 isthesetofverticesofQ.TheA-moduleMhasadimensionvectordim(M):= (dim(e i M)) i ∈ N Q 0 whichrefinesthenotionofdimension.TheGrassmannianGr e (M,A) dimensionvectoresuchthat i e i =e.Henceitisnaturaltostudythegeometric splitsintoadisjointunionofclos P edsubvarieties Gr e (M,A)ofsubmodulesofMwithgiven propertiesoftheso-calledquiverGrassmannian Gr e (M):= Gr e (M,A). ThequiverGrassmanniancanbefoundin[8,22]inconnectionwiththedimensionof spacesofmorphimsofkQ-modules.ItplaysanimportantroleinFomin-Zelevinsky’stheory ofclusteralgebras,[10].Indeed,asprovedin[3]and[5],Eulercharacteristicsχ c ( Gr e (M)) ofquiverGrassmanniansdefinesetsofgeneratorsofclusteralgebrasintheacyclictype. Tobemoreprecise,considertheringofLaurentpolynomial Z [x i ± ,i ∈ Q 0 ]andforany (f.d.)kQ-moduleMwithdimensionvectorm=(m i ),set: X M :=x −m i χ c ( Gr e (M))x m t ( h ) −e t ( h ) x e s ( h ) , YXY i s(h)t(h) i∈Q 0 e h∈Q 1 whereQ 1 isthesetofarrowsofQ,ands(h),t(h)arerespectivelythesourceandthe targetofthearrowh.Then,itisknownthatasetofgeneratorsoftheclusteralgebra isgivenby { X M } whereMrunsoverthesetof exceptionalmodules , i.e. moduleswith noself-extension.Thesegeneratorsarecalledclustervariables.Moreover,theset { X M } , whereMrunsoverthesetoff.d.modulesis(atleastconjecturally)a Z -baseofthecluster algebrawhichisananalogueofLusztig’sdualsemicanonicalbase,[6]. TheaimofthepaperistostudygeometricpropertiesofquiverGrassmannians,andin particulartounderstandtheirEulercharacteristics. 1