Bases of representations of type A affine Lie algebras via q(4)

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

4IGORB.FRENKELANDALISTAIRSAVAGE

Fortwointegersk′≤k,de neV∞(k′,k)∈Vtobethevectorspacewithbasis{er|k′≤r≤k}.Werequirethaterhasdegreer∈I.Letx∞(k′,k)∈EV∞(k′,k), bede nedbyx∞(k′,k):er→er 1fork′≤r≤k,whereek′ 1=0.Itisclearthat(V∞(k′,k),x∞(k′,k))isanindecomposablerepresentationofourquiver.Conversely,anyindecomposable nite-dimensionalrepresentation(V,x)ofourquiverisisomorphictosome(V∞(k′,k),x∞(k′,k)). ∞bethesetofallLetZ∞bethesetofallpairs(k′≤k)ofintegersandletZ

functionsZ∞→Nwith nitesupport.

ItiseasytoseethatforV∈V,thesetofGV-orbitsinEV, isnaturallyindexed ∞ofZ ∞consistingofthosef∈Z ∞suchthatbythesubsetZV

f(k′,k)=dimVi

k′≤i≤k

foralli∈I.Herethesumisoverallk′≤ksuchthatk′≤i≤k.Correspondingtoagivenfistheorbitconsistingofallrepresentationsisomorphictoasumoftheindecomposablerepresentationsx∞(k′,k),eachoccuringwithmultiplicityf(k′,k). ∞.LetCfbetheconormalDenotebyOftheGV-orbitcorrespondingtof∈ZV¯fbeitsclosure.Wethenhavethefollowingproposition.bundletoOfandletC

¯fisaone-to-onecorrespondencebetweentheProposition1.1.2.Themapf→C ∞andthesetofirreduciblecomponentsofΛV.setZV

Proof.ThisfollowsimmediatelyfromProposition1.1.1.

(1)(1) 1.2.TypeAn.Letgbethea neLiealgebraoftypeAn,thatis,theLiealgebra

generatedbythesetofelementsEk,Fk,Hk(k=0,1,...,n)anddsatisfyingthefollowingrelations:

[Ek,Fl]=δklHk,[Hk,El]=aklEl,[Hk,Fl]= aklFl,

fork=l.[d,Ek]=δk0Ek,(adEk)1 aklEl=0,

Here

akl=2δ(k,l) δ(k,l+1) δ(k,l 1),

whereδ(k,l)=1ifk≡lmod(n+1)andisequaltozerootherwise.

LetI=Z/(n+1)Zbethesetofverticesofagraphwiththesetoforientededgesgivenby

H={i→j|i,j∈I,i j=1}∪{i←j|i,j∈I,i j=1}.

Fortwointegersk′≤k,de neV(k′,k)∈Vtobethevectorspacewithbasis{er|k′≤r≤k}.Werequirethaterhasdegreei∈Iwherer≡i(modn+1).Letx(k′,k)∈EV(k′,k), bede nedbyx(k′,k):er→er 1fork′≤r≤k,whereek′ 1=0.Itisclearthat(V(k′,k),x(k′,k))isanindecomposablerepresentationofourquiverandthatx(k′,k)isnilpotent.Also,theisomorphismclassofthisrepresentationdoesnotchangewhenk′andkaresimultaneouslytranslatedbyamultipleofn+1.Conversely,anyindecomposable nite-dimensionalrepresentation(V,x)ofourquiver,withxnilpotent,isisomorphictosome(V(k′,k),x(k′,k))wherek′andkareuniquelyde neduptoasimultaneoustranslationbyamultipleofn+1.

LetZbethesetofallpairs(k′≤k)ofintegersde neduptosimultaneous bethesetofallfunctionsZ→Nwithtranslationbyamultipleofn+1andletZ

nitesupport.[d,Fk]= δk0Fk,(adFk)1 aklFl=0

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