Bases of representations of type A affine Lie algebras via q(9)

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,QUIVERVARIETIESANDSTATISTICALMECHANICS9

1ek’111ek’+1ek’+2ek’+3e1k1e1k+1ek+211ekek 1k’2k’+12k’ 2k’ 1k’111k’+1k’+211kk 122

Figure1.Ifxr+1←r(e2r)=0forsomer,thecommutativityof′′thisdiagramforcesk2<k1andk2<k1.Verticesrepresentthe

spansoftheindicatedvectors.Thosealignedverticallylieinthe

sameVi.Thearrowsindicatetheactionoftheobviouscomponent

ofx

.

Figure2.TheirreduciblecomponentsofL(v,w0)areenumer-

atedbyYoungdiagrams.ThetoplineistheDynkingraphoftype

A∞.Theotherhorizontallinesrepresentx∞(k′,k)wherek′and

karethepositionsoftheleftmostandrightmostvertices.

Now,letxlieintheconormalbundletothepoint

(4.1.1)s i=1′′′′x(ki,ki+li 1)∈E⊕s.i=1V∞(ki,ki+li 1),

′′Wecanassume(byreorderingtheindicesifnecessary)thatk1≥k2≥···≥′iks.Now,bytheabovearguments,xr+1←r(er)mustbealinearcombinationof

{ejr+1}j<i.Thus

e1′∈kerxk′→k′ 1∩kerxk′+1←k′.k11111

′Bythestabilitycondition,wemustthenhavek1=0andtherecanbenootherierinkerxr→r 1∩kerxr+1←rforanyr.Now,bytheaboveconsiderations,e2′isk2′′2′→k′ 1∩kerxk′+1←k′unlessk+1=k′←k′(e′)isanon-zeroinkerxk221andxk1k22222

1′′multipleofek′.Continuinginthismanner,weseethatwemusthaveki+1+1=ki

i+1′←k′andxki(e)=ciei′=0for1≤i≤s 1.Thenbytheabovewemusthave′kikii+1+1′ki+1<kifor1≤i≤s 1.Settingli=ki ki+1thetheoremfollows. 1

TheYoungdiagramsenumeratingtheirreduciblecomponentsofL(v,w0)canbevisualizedasinFigure2.NotethattheverticesinourdiagramcorrespondtotheboxesintheclassicalYoungdiagram,andourarrowsintersecttheclassicaldiagramedges.

FortheleveloneA∞case,itisrelativelyeasytocomputethegeometricactionofthegeneratorsEkandFkofg.We rstnotethatforeveryv,L(v,w0)iseitheremptyorisapoint.ItfollowsthateachXYisequaltoL(v,w0)forsomeuniquevwhichwewilldenotevY.