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,QUIVERVARIETIESANDSTATISTICALMECHANICS11Proof.TheargumentisexactlyanalogoustothatusedintheproofofTheo-rem4.1.1.Weneedonlynotethatapointintheconormalbundletotheorbitthroughthepoint
(4.2.1)s i=1′′′′x(ki,ki+li 1)∈E⊕si=1V(ki,ki+li 1),
liesinΛV(v,w0)ifandonlyifli>li+nforalli=1,...,s(li=0fori>s)bytheaperiodicitycondition.
NotethatNakajima’sconstructionyieldsanactionoftheLiealgebraonthebasis{gXY}Y∈Ynofthebasicrepresentation.However,thisactionisnotasstraightfor-wardtocomputeasintheA∞caseandwillbeconsideredinafuturework.
4.3.gln+1Case.Wede neAYforY∈YasinSection4.1.
{(k′,k)|f(k′,k)=1}=AY
forsomeY∈Yandf(k′,k)=0for(k′,k)∈AY(uptosimultaneoustranslationofk′andkbyn+1).DenotethecomponentcorrespondingtosuchanfbyXY.Thus,Y XYisanatural1-1correspondencebetweenthesetYandtheirreducible (v,w0).componentsof∪vL
Proof.TheargumentisexactlyanalogoustothatusedintheproofofTheo-rem4.1.1.
n+1isnotaKac-MoodyalgebraweneedtoAsnotedinSection1.3,sincegl
modifyNakajima’sconstructionofhighestweightrepresentations.Notethatfor n+1andsl n+1isthesameHeisenbergalgebragl 1.anyn,thedi erencebetweengl
TherepresentationsofHeisenbergalgebrasinthecontextofgeometricrepresenta-tiontheorywere rstconstructedbyGrojnowski[G]andNakajima[N2](see[N4]forareview).However,itisnotobvioushowtoadaptthisrepresentationtheory (v,w0),obtainingthedesiredcommutationrelationstothenewquivervarietiesL n+1.Thisproblemwillbeconsideredinafuturework.withthegeneratorsofsl
5.ArbitraryLevelRepresentations
5.1.TypeA∞.We rstrecallsomede nitionsfrom[DJKMO2].AMayadiagramisabijectionm:Z→Zsuchthat(m(j))j<0and(m(j))j≥0arebothincreasing.ForeachMayadiagramthereexistsauniqueγ∈Zsuchthatm(j) j=γfor|j| 0.Thisγiscalledthechargeofm.WedenotethesetofMayadiagramsofchargeγbyM[γ].Form∈M[γ]welet
m[r]=(m(j)+r)j∈Z∈M[γ+r].
WecanvisualizeaMayadiagrambyaYoungdiagram.Consideralatticeontherighthalfplanewithlatticepoints{(i,j)∈Z2|i≥0}.Eachedgeonthelatticeisoriented,startingat(i,j)andendingat(i+1,j)or(i,j+1)andisnumberedbytheintegeri+j.ApathonthelatticeisamapefromZtothesetofedgesonthelatticesuchthate(j)hasnumberjandtheendingsiteofe(j)isthestartingsiteofe(j+1).ToeachMayadiagramofchargeγ,weassociatetheuniquepathsatisfyingthefollowingconditions.
(1)Forj 0,e(j)istheedgefrom(0,j)to(0,j+1),
(2)Theedgee(m(j))isvertical(resp.horizontal)ifj<0(resp.j≥0). (v,w0)arepreciselythoseXfTheorem4.3.1.TheirreduciblecomponentsofL Vsuchthatwheref∈Z