Bases of representations of type A affine Lie algebras via q(13)

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,QUIVERVARIETIESANDSTATISTICALMECHANICS13

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Figure4.Thestringsassociatedtosomex∈EV, .Thetopline

istheDynkindiagramoftypeA∞.

andtheirreduciblecomponentsof∪vL(v,w).

Proof.RecallthatirreduciblecomponentsofL(v,w)aretheclosuresoftheGV-orbits(orisomorphismclasses)inEV, andthatthereisarepresentativeofeachorbitoftheform x∞(k′,k)(5.1.1)x=

(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

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