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,QUIVERVARIETIESANDSTATISTICALMECHANICS5ItiseasytoseethatforV∈V,thesetofGV-orbitsonthesetofnilpotent VofZ consistingofthoseelementsinEV, isnaturallyindexedbythesubsetZ suchthatf∈Z f(k′,k)#{r|k′≤r≤k,r≡i(modn+1)}=dimVi
k′≤k
foralli∈I.Herethesumistakenoverallk′≤kuptosimultaneoustranslationbyamultipleofn+1.Correspondingtoagivenfistheorbitconsistingofallrepresentationsisomorphictoasumoftheindecomposablerepresentationsx(k′,k),eachoccuringwithmultiplicityf(k′,k).DenotebyOftheGV-orbitcorresponding V.tof∈Z Visaperiodicifforanyk′≤k,notallf(k′,k),f(k′+1,k+1),Wesaythatf∈Z V,letCfbetheconormal...,f(k′+n,k+n)aregreaterthanzero.Foranyf∈Z¯fbeitsclosure.bundleofOfandletC
V.ThefollowingtwoconditionsareProposition1.2.1([L1,15.5]).Letf∈Z
equivalent.
(1)Cfconsistsentirelyofnilpotentelements.
(2)fisaperiodic.
¯fisa1-1correspondencebetweenProposition1.2.2([L1,15.6]).Themapf→C VandthesetofirreduciblecomponentsofΛV.thesetofaperiodicelementsinZ
′′′Proposition1.2.3([L1,12.8]).Letx′∈EV, andx′′∈EV, ¯.Thenψi(x+x)=
0foralli∈Iifandonlyifx′′isorthogonalwithrespectto , tothetangentspacetotheGV-orbitofx′,regardedasavectorsubspaceofEV, .
1.3.gln+1Case.Sincegln+1isnotaKac-Moodyalgebrainastrictsense,thiscaseisnotcoveredbyLusztig’stheoryandrequirescertainmodi cations.Wepreservethenotationoftheprevioussubsection.
Vbethesetofallelementsx=x′+x′′,wherex′∈EV, De nition1.3.1.LetΛ′andx′′∈EV, ¯,suchthatxisnilpotentandψi(x)=0foralli∈I.
V,wedenotebyOfthecorrespondingGV-orbitandbyCfitsForanyf∈Z
conormalbundle.
V.ThenProposition1.3.2.Letf∈Z
V,and(1)CfconsistsentirelyofelementsofΛ VistheunionofC¯fforallf∈Z V.(2)Λ
Proof.ThisfollowsfromProposition1.2.3.
¯fisa1-1correspondencebetweenthesetZ VProposition1.3.3.Themapf→C
andthesetofirreduciblecomponentsofΛV.
Proof.ThisfollowseasilysincetheconormalbundlesCfareirreducibleofthesamedimension.
2.Nakajima’sQuiverVarieties
Weintroducehereadescriptionofthequivervarieties rstpresentedin[N1]in
(1)thecaseoftypesA∞andAn.
De nition2.0.1([N1]).Forv,w∈ZI≥0,chooseI-gradedvectorspacesVandWofgradeddimensionsvandwrespectively.Thende ne Λ≡Λ(v,w)=ΛV×Hom(Vi,Wi).
i∈I