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

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

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