Bases of representations of type A affine Lie algebras via q(14)

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

14IGORB.FRENKELANDALISTAIRSAVAGE

wesetmi=(Yi,γi).Byconstruction,thelengthofanystringassociatedtoγiisgreaterthanthelengthofastringwiththesameleftendpointassociatedtoγjforj>i.Thisimmediatelyyieldstheconditionm1≤···≤ml.Ourconstructionthusgivesustheone-to-onecorrespondenceassertedintheTheorem.

NotethattheenumerationoftheirreduciblecomponentsgiveninTheorem5.1.2matchesthatofProposition4.6of[DJKMO2].

5.2.TypeAn.WenowconsiderthecasewheregisoftypeAn.ForanelementM=(m1,...,ml)∈M[Λ],letRMbetheset(withmultiplicity)ofpairs(i,li)whereliisthelengthofarowwithtopedgehavingy-coordinateibelongingtooneofthemj.WesaythatMisn-reducedif

{(k+i,l)|0≤i≤n} RM

forallkandl.

De nefMforM∈M[Λ]asintheprevioussubsection(exceptthatnowourpairsarede nedonlyuptosimultaneoustranslationbyn+1).

Theorem5.2.1.TheirreduciblecomponentsofL(v,w)arepreciselythoseXfwheref=fMforsomen-reducedM∈M[Λ].DenotethecomponentXfMbyXM.ThenM XMisanaturalone-to-onecorrespondencebetweentheset

{(m1,...,ml)∈M[Λ]|m1≤···≤ml≤m1[n+1],Misn-reduced}

andtheirreduciblecomponentsof∪vL(v,w).

Proof.EachirreduciblecomponentcorrespondstoasetofstringsasintheproofofTheorem5.1.2withtheaddedconditionthatwecannothaven+1strings,eachofthesamelength,whoseleftendpointsarethen+1verticesofourquiver.Thatis,wemusthavethatMisn-reduced.NotethattheprocessdescribedintheproofofTheorem5.1.2yieldsmi=(Yi,γi)satisfyingm1≤···≤ml≤m1[n+1]asdesired.TheTheoremfollows.

Again,asnotedinSection4.2,Nakajima’sconstructionyieldsanactionoftheLiealgebraonthebases{gXM}oftheirreduciblerepresentationsinboththeA∞

(1)andAncaseswhichismoredi culttodirectlycomputethaninthelevel1A∞case.

5.3.gln+1Case.(1)(1)

(v,w)arepreciselythoseXfTheorem5.3.1.TheirreduciblecomponentsofL

wheref=fMforsomeM∈M[Λ].DenotethecomponentXfMbyXM.ThenM XMisanaturalone-to-onecorrespondencebetweentheset

{(m1,...,ml)∈M[Λ]|m1≤···≤ml≤m1[n+1]}

(v,w).andtheirreduciblecomponentsof∪vL

Proof.TheargumentisthesameastheproofofTheorem5.2.1exceptthatwedonothavetheaperiodicityconditionandthusdonotrequirethatMisn-reduced.

(v,w)givenNotethattheenumerationoftheirreduciblecomponentsof∪vL

byTheorem5.3.1isthesameasthatgivenbyProposition4.7of[DJKMO2]fora spanningsetofthedualtotheirreduciblehighestweightrepresentationofgln+1.In

n+1ordertoextendthegeometricconstructionofhighestweightrepresentationsofsl togln+1foranarbitrarylevel,onewouldneedarepresentationoftheHeisenberg

algebraasdiscussedinSection4.3.Hereonemightusetheconstructionofthe

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