Bases of representations of type A affine Lie algebras via q(18)

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

18IGORB.FRENKELANDALISTAIRSAVAGE

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Figure5.Removingan(n+1)×lsquarefromaMayadiagram

(heren=2andl=4).Noticethattheenumerationofthehori-

zontaledgesdoesnotchangemod(n+1).

Recallthede nitionofRMgiveninSection5.2.ForM,M′∈M[Λ],wesaythatMisann-reductionofM′ifRMisobtainedfromRM′bytheremovalofsetsoftheform{(k+i,l)|0≤i≤n}forsomekandl.

′Proposition6.2.1.SupposeM=(m1,...,ml)isann-reductionofM′=(m′1,...,ml)′′′andm1≤···≤ml≤m1[n+1],m′1≤···≤ml≤m1[n+1].ThenMandMare

liftsofthesamepathandM≥M′.

Proof.RecalltheconstructionintheproofofTheorem5.1.2.Notethatchoosingarbitrarystringsinsteadofthelongeststringateachstepwillnotchangethevalues′′oftherighthandsideof(6.2.2)(foranyk).Thus,letusformM′′=(m′′1,...,ml)∈M[Λ]fromthesamestringscomprisingM′butwhereoneofthem′′icontainstheentiresetofstringsoftheform{(k+i,l)|0≤i≤n}whichisremovedfromRM′toobtainRM.Now,removingthissetofstringsfromM′′simplyamountstoremovingthissetfromm′′i.Butthisjustcutsan(n+1)×lsquareoutoftheMayadiagram′′miandshiftsthepartofthediagrambelowthecutupn+1units.SeeFigure5.

Since µ+n+1= µ,thevaluesoftherighthandsidesof(6.2.2)forM′andM′′arethesame.However,MissimplyobtainedfromM′′byapplyingtheprocedureofTheorem5.1.2tothestringsofM′′andasmentionedabove,thisdoesnotchangethevalueoftherighthandsidesof(6.2.2).ThusMandM′areliftsofthesamepath.

ToshowthatM≥M′,notethatbytheconstructionintheproofofTheo-rem5.1.2,MisuniquelydeterminedbyRM.Now,weobtainRMfromRM′byremovingasetoftheform{(k+i,l)|0≤i≤n}forsomekandl.Thus,ateachstageinourconstructionofM,wechoseastringoflengthlessthanorequaltothestringchosenintheconstructionofM′.ThuswehavethatM≥M′. Proposition6.2.2([DJKMO2]).ForeachΛ-pathηthereexistsauniquehighestliftMofηsuchthatM≥M′foranyliftM′ofη.

Corollary6.2.3.Theset

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

ispreciselythesetofhighestliftsofpathsinP(Λ).

LetMηbethen-reducedelementofM[Λ]correspondingtoη∈P(Λ)andletg

(1)bethea neLiealgebraoftypeAn.De ne

P(Λ)µ={η∈M[Λ]|λη=µ}.

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