Bases of representations of type A affine Lie algebras via q(5)

(k′≤k)∈K

forsome nitesetofpairsK.Bypicturingx∞(k′,k)asthestringofverticesk′,k′+1,...,k,wecanrepresentsuchanxbyasetof nitestringsofverticescorrespondingtothevariousx∞(k′,k)appearingin(5.1.1).Wecallthenumberofverticesinastringitslength.EachvertexofastringrepresentsabasisvectorofVwithdegreegivenbythelocationofthevertex.Theactionofxmapseachofthesebasisvectorstothebasisvectorcorrespondingtothenext(onelower)vertexinthestring(ortozeroifnosuchvertexexists).SeeFigure4.

ItisthenastraightforwardextensionoftheproofofTheorem4.1.1thattheallowablesetsofstringsarepreciselythosethatcanbegroupedintosubsets,oneforeachγi,suchthatthesubsetcorrespondingtoγi,whenorderedbydecreasingleftmostvertex,hasweaklydecreasinglengths,the rstleftmostvertexisγiandtheleftmostverticesdecreasebyoneaswemovethroughthesubsetinorder(byleftmost,wemeanthevertexwiththesmallestindex).Thisispreciselythe rstclaimoftheTheorem.

Itiseasytoseethatmanydi erentM∈M[Λ]maycorrespondtothesameirreduciblecomponent.Forexample,forΛ=Λ 1+Λ1,both

M=(([3,2,1], 1),([4,3,2,1],1)),and

M′=(([2,1], 1),([4,3,3,2,1],1))

belongtoM[Λ]andcorrespondtothesetofstringsshowninFigure4(andhencetothesameirreduciblecomponent).However,wecanassociateauniqueM∈M[Λ]toeachsetofstringsdescribedaboveasfollows.Associatetoγ1thelongeststringwithleftmostvertexγ1andremovethisstringfromtheset.Nowdothesameforγ2,etc.Afterwehaveassociatedastringtoγl,westartagainatγ1butthistimeweselectthelongeststringwithleftmostvertexγ1 1andsoforth.Ifatanypoint,thereisnostringtoassociatewithagivenγi,weremovethisγifromfurthersteps.Inthiswayweassociatetoeachγiasequenceofstringsofweaklydecreasinglength(byourconditiononthepossiblesetsofstrings)withleftmostverticesdecreasingbyone.ThelengthsofthestringsassociatedtoγigiveaYoungdiagramYiand

We relate two apparently different bases in the representations of affine Lie algebras of type A: one arising from statistical mechanics, the other from gauge theory. We show that the two are governed by the same combinatorics and therefore can be viewed a

14IGORB.FRENKELANDALISTAIRSAVAGE

wesetmi=(Yi,γi).Byconstruction,thelengthofanystringassociatedtoγiisgreaterthanthelengthofastringwiththesameleftendpointassociatedtoγjforj>i.Thisimmediatelyyieldstheconditionm1≤···≤ml.Ourconstructionthusgivesustheone-to-onecorrespondenceassertedintheTheorem.

NotethattheenumerationoftheirreduciblecomponentsgiveninTheorem5.1.2matchesthatofProposition4.6of[DJKMO2].

5.2.TypeAn.WenowconsiderthecasewheregisoftypeAn.ForanelementM=(m1,...,ml)∈M[Λ],letRMbetheset(withmultiplicity)ofpairs(i,li)whereliisthelengthofarowwithtopedgehavingy-coordinateibelongingtooneofthemj.WesaythatMisn-reducedif

{(k+i,l)|0≤i≤n} RM

forallkandl.

De nefMforM∈M[Λ]asintheprevioussubsection(exceptthatnowourpairsarede nedonlyuptosimultaneoustranslationbyn+1).

Theorem5.2.1.TheirreduciblecomponentsofL(v,w)arepreciselythoseXfwheref=fMforsomen-reducedM∈M[Λ].DenotethecomponentXfMbyXM.ThenM XMisanaturalone-to-onecorrespondencebetweentheset

{(m1,...,ml)∈M[Λ]|m1≤···≤ml≤m1[n+1],Misn-reduced}

andtheirreduciblecomponentsof∪vL(v,w).

Proof.EachirreduciblecomponentcorrespondstoasetofstringsasintheproofofTheorem5.1.2withtheaddedconditionthatwecannothaven+1strings,eachofthesamelength,whoseleftendpointsarethen+1verticesofourquiver.Thatis,wemusthavethatMisn-reduced.NotethattheprocessdescribedintheproofofTheorem5.1.2yieldsmi=(Yi,γi)satisfyingm1≤···≤ml≤m1[n+1]asdesired.TheTheoremfollows.

Again,asnotedinSection4.2,Nakajima’sconstructionyieldsanactionoftheLiealgebraonthebases{gXM}oftheirreduciblerepresentationsinboththeA∞

(1)andAncaseswhichismoredi culttodirectlycomputethaninthelevel1A∞case.

5.3.gln+1Case.(1)(1)

(v,w)arepreciselythoseXfTheorem5.3.1.TheirreduciblecomponentsofL

wheref=fMforsomeM∈M[Λ].DenotethecomponentXfMbyXM.ThenM XMisanaturalone-to-onecorrespondencebetweentheset

{(m1,...,ml)∈M[Λ]|m1≤···≤ml≤m1[n+1]}

(v,w).andtheirreduciblecomponentsof∪vL

Proof.TheargumentisthesameastheproofofTheorem5.2.1exceptthatwedonothavetheaperiodicityconditionandthusdonotrequirethatMisn-reduced.

(v,w)givenNotethattheenumerationoftheirreduciblecomponentsof∪vL

byTheorem5.3.1isthesameasthatgivenbyProposition4.7of[DJKMO2]fora spanningsetofthedualtotheirreduciblehighestweightrepresentationofgln+1.In

n+1ordertoextendthegeometricconstructionofhighestweightrepresentationsofsl togln+1foranarbitrarylevel,onewouldneedarepresentationoftheHeisenberg

algebraasdiscussedinSection4.3.Hereonemightusetheconstructionofthe

We relate two apparently different bases in the representations of affine Lie algebras of type A: one arising from statistical mechanics, the other from gauge theory. We show that the two are governed by the same combinatorics and therefore can be viewed a

AFFINELIEALGEBRAS,QUIVERVARIETIESANDSTATISTICALMECHANICS15HeisenbergalgebrabyBaranovsky[B]thatgeneralizestheGrojnowski/Nakajimaconstructiontohigherlevels.

Remark5.3.2.OnecanalsogiveageometricinterpretationofthefullFockspaceof[DJKMO2]withbasisindexedbyM[Λ]viathe“smooth”Ul-instantonmodulispace rM(r,l)whichhasthesamegeneratingfunctionforcohomology(seee.g.

[N4],Chapter5)asthefullFockspacewiththebasisM[Λ].ThetypesA(1)norA∞arere ectedintherespectiveactionofthegroupsZ/(n+1)ZorC onthemodulispace,andγ1,...,γlisthesetofone-dimensionalrepresentationsofthesegroupsthatdeterminethisaction.

6.AComparisonWithThePathSpaceRepresentation

Theauthorsof[DJKMO1]constructedthebasicrepresentationofAn(1)onthespaceofpaths.In[DJKMO2],thispathrealizationisgeneralizedtoarbitrarylevel.WenowcomparethegeometricpresentationL(v,w)withtheirs.Wewillslightlymodifythede nitionsof[DJKMO1]toagreewiththemoregeneralde nitionsof

[DJKMO2].

6.1.TheLevelOneCase.Abasicpat …… 此处隐藏：5674字，全部文档内容请下载后查看。喜欢就下载吧 ……