Positivity for Kac polynomials and DT-invariants of quivers

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1、PositivityofKacpolynomialsandDT-invariantsforquiversTamasHausel´EmmanuelLetellierEPFLausanneUniversit´edeCaentamas.hausel@epfl.chletellier.emmanuel@math.unicaen.frFernandoRodriguez-VillegasUniversityofTexasatAustinvillegas@math.utexas.eduApril12,2012AbstractWegiveacohomologicalinte

2、rpretationofboththeKacpolynomialandtherefinedDonaldson-Thomas-invariantsofquivers.ThisinterpretationyieldsaproofofaconjectureofKacfrom1982andgivesanewperspectiveonrecentworkofKontsevich–Soibelman.Thisisachievedbycomputing,viaanarithmeticFouriertransform,thedimensionsoftheisoytpicalc

3、omponentsofthecohomologyofassociatedNakajimaquivervarietiesundertheactionofaWeylgroup.ThegeneratingfunctionofthecorrespondingPoincar´epolynomialsisanextensionofHua'sformulaforKacpolynomialsofquiversinvolvingHall-Littlewoodsymmetricfunctions.Theresultingformulaecontainawiderangeofin

4、formationonthegeometryofthequivervarieties.1ThemainresultsarXiv:1204.2375v1[math.RT]11Apr2012LetΓ=(I,Ω)beaquiver:thatis,anorientedgraphonafinitesetI={1,...,r}withΩafinitemultisetoforientededges.Inhisstudyoftherepresentationtheoryofquivers,Kac[17]introducedAv(q),thenumberofisomorphism

5、classesofabsolutelyindecomposablerepresentationsofΓoverthefinitefieldFqofdimensionv=(v1,...,vr)andshowedtheyarepolynomialsinq.WecallAv(q)theKacpolynomialforΓandv.FollowingideasofKac[17],Hua[16]provedthefollowinggeneratingfunctionidentity:QXXqhπi,πjivi→j∈Ω

6、π

7、Av(q)T=(q−1)

8、LogQhπi,πiiQQmk(πi)−jT,(1.1)v∈Zr{0}π=(π1,...,πr)∈Pri∈Iqkj=1(1−q)≥0wherePdenotesthesetofpartitionsofallpositiveintegers,Logistheplethysticlogarithm(see[14,§2.3.3]),h,iisthepairingonpartitionsdefinedbyXhλ,µi=min(i,j)mi(λ)mj(µ)i,j12withm(λ)themultiplicityofthepartjinthepartitionλ,

9、Tv:=Tv1···TvrforsomevariablesTandfinallyj1ri

10、π

11、:=(

12、π1

13、···

14、πr

15、)UsingsuchgeneratingfunctionsKac[17]provedthatinfactAv(q)hasintegercoefficientsandformulatedtwomainconjectures.First,heconjecturedthatforquiverswithnoloopstheconstanttermAv(0)equalsthemultiplicityoftherootvinthecorresponding

16、Kac-Moodyalgebra.Theproofo