We use the idea of generic extensions to investigate the correspondence between the isomorphism classes of nilpotent representations of a cyclic quiver and the orbits in the corresponding representation varieties. We endow the set $\cal M$ of such isoclass

MONOMIALBASESFORQUANTUMAFFINEsln

arXiv:math/0307257v1 [math.RA] 18 Jul 2003BANGMINGDENGANDJIEDUAbstract.Weusetheideaofgenericextensionstoinvestigatethecorrespondencebe-tweentheisomorphismclassesofnilpotentrepresentationsofacyclicquiverandtheorbitsinthecorrespondingrepresentationvarieties.WeendowthesetMofsuchisoclasseswithamonoidstructureandidentifythesubmonoidMcgeneratedbysimplemodules.Ontheotherhand,weusethepartialorderingontheorbits(i.e.,theBruhat-Chevalleytypeordering)toinduceaposetstructureonManddescribetheposetidealsgeneratedbyanelementofthesubmonoidMcintermsoftheexistenceofacertaincompositionseriesofthecorrespondingmodule.Asapplicationsoftheseresults,wegeneralizesomeresultsofRingelinvolvingspecialwordstoresultswithnorestrictiononwordsandobtainasystematicdescriptionofmanymonomialbasesforanygivenquantuma nesln.1.IntroductionLetUbeaquantumgroupoverQ(v)associatedtoaCartandatum(I, )inthesenseof[9,1.1],andletEi,Fi,Ki±1(i∈I)beitsgenerators.ThenallmonomialsintheEi’s,Fi’sandKi±1’sspanU.ItisnaturaltoaskwhichmonomialsformbasesforU.SinceUadmitsatriangulardecompositionU=U U0U+whereU+(resp.U ,U0)isthesubalgebra generatedbytheEi’s(resp.Fi’s,Ki±1’s),anda10anwitha=themonomialsKa=K1···Kniaii∈ZIformabasisforU,itwouldbeinterestingtoknowmonomialbasesforU+(andhenceforU ).Moreprecisely,let bethesetofwordsonthealphabetI.Foreachwordw=i1···im∈ ,letEw=Ei1···Eim,Fw=Fi1···Fim.Weareinterestedin ndingsubsets ′ suchthattheset{Ew}w∈ ′formsabasisforU+.Inthecasewhere(I, )isofsimply-laced nitetype,Lusztigintroducedcertainmonomial

basesforU+in[7,7.8]relativetoa xedreducedexpressionofthelongestwordinthecorrespondingWeylgroup.Interestingapplicationsofmonomialbasescanbefoundin,e.g.,[15],[2],[11]and[3].In[3]itisprovedthatsome(integral)monomialbasesforaquantumglngiverisetomonomialbasesforq-SchuralgebrasandHeckealgebras(see,e.g.,

[3,7.2,9.4]).

Intheinvestigation[13]ofrealizingthegenericcompositionalgebraofacyclicquiverasthe+partofaquantuma nesln,Ringelconstructedcertainmonomialbasesoversome