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

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

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

10IGORB.FRENKELANDALISTAIRSAVAGE

Lemma4.1.2.ThefunctiongXYcorrespondingtotheirreduciblecomponentXYwhereY∈Yissimply1XY,thefunctiononXYwithconstantvalueone.

Proof.ThisisobvioussinceXYisapoint. Proposition4.1.3.OnehasFk1XY=1XY′wherevY′=vY+ekifsuchaY′existsandFk1XY=0otherwise.Also,Ek1XY=1XY′′wherevY′′=vY ekifsuchaY′′existsandEk1XY=0otherwise.

Proof.Itisclearfromthede nitionsthatFk1XY=c11XY′andEk1XY=c21XY′′forsomeconstantsc1andc2ifY′andY′′existasdescribedaboveandthattheseactionsarezerootherwise.Wesimplyhavetocomputetheconstantsc1andc2.Now,

1XY(x)Fk1XY(x)=(π2)!π1

=χ({S|Sisx-stable,x|S∈XY})

=χ(pt)

=1

ifx∈XY′wherevY′=vY+ekandzerootherwise.ThefactthattheabovesetissimplyapointfollowsfromthefactthatSkmustbethesumoftheimagesofxhsuchthatin(h)=k.Thusc1=1asdesired.

NotethatifthereexistsaY′suchthatvY′=vY+ekthentherecannotexistaY′′suchthatvY′′=vY ekandviceversa.ThereforeifsuchaY′′exists,Fk1XY=0andso

Hk1XY=[Ek,Fk]1XY= FkEk1XY.

OnecaneasilycheckthatHk1XY= 1XYifaY′′existsasdescribedaboveandthusFkEk1XY=1XY.Itthenfollowsfromtheabovethatwemusthavec2=1.

TheaboveactionofthetypeA∞LiealgebrainthespacespannedbyabasisindexedbyYoungdiagramsiswellknowninapurelyalgebraiccontext(seee.g.

[JM]).

Remark4.1.4.Alltheresultsofthissectioncanberepeatedwithminormodi ca-tionsforfundamentalrepresentationsof nite-dimensionalLiealgebrasoftypeAn.Inthiscase,thebasesoffundamentalrepresentationswillbeenumeratedbyYoungdiagramsofsizeboundedbyanm×(n+1 m)rectangle,wherem=1,2,...,nistheindexofthefundamentalrepresentation.NotethatthesameYoungdiagramsalsonaturallyenumeratetheSchubertcellsoftheGrassmanniansGr(m,n+1)fortypeAnorthesemi-in niteGrassmannianfortypeA∞.

4.2.TypeAn.LetYnbethesetofallYoungdiagrams[l1,...,ls]satisfyingli>li+nforalli=1,...,s(lj=0forj>s).ForY=[l1,...,ls]∈Yn,letAYbetheset{(1 i,li i)|1≤i≤s}.

Theorem4.2.1.TheirreduciblecomponentsofL(v,w0)arepreciselythoseXf Vsuchthatwheref∈Z

{(k′,k)|f(k′,k)=1}=AY

forsomeY∈Ynandf(k′,k)=0for(k′,k)∈AY(uptosimultaneoustranslationofk′andkbyn+1).DenotethecomponentcorrespondingtosuchanfbyXY.Thus,Y XYisanatural1-1correspondencebetweenthesetYnandtheirreduciblecomponentsof∪vL(v,w0).(1)

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,QUIVERVARIETIESANDSTATISTICALMECHANICS11Proof.TheargumentisexactlyanalogoustothatusedintheproofofTheo-rem4.1.1.Weneedonlynotethatapointintheconormalbundletotheorbitthroughthepoint

(4.2.1)s i=1′′′′x(ki,ki+li 1)∈E⊕si=1V(ki,ki+li 1),

liesinΛV(v,w0)ifandonlyifli>li+nforalli=1,...,s(li=0fori>s)bytheaperiodicitycondition.

NotethatNakajima’sconstructionyieldsanactionoftheLiealgebraonthebasis{gXY}Y∈Ynofthebasicrepresentation.However,thisactionisnotasstraightfor-wardtocomputeasintheA∞caseandwillbeconsideredinafuturework.

4.3.gln+1Case.Wede neAYforY∈YasinSection4.1.

{(k′,k)|f(k′,k)=1}=AY

forsomeY∈Yandf(k′,k)=0for(k′,k)∈AY(uptosimultaneoustranslationofk′andkbyn+1).DenotethecomponentcorrespondingtosuchanfbyXY.Thus,Y XYisanatural1-1correspondencebetweenthesetYandtheirreducible (v,w0).componentsof∪vL

Proof.TheargumentisexactlyanalogoustothatusedintheproofofTheo-rem4.1.1.

n+1isnotaKac-MoodyalgebraweneedtoAsnotedinSection1.3,sincegl

modifyNakajima’sconstructionofhighestweightrepresentations.Notethatfor n+1andsl n+1isthesameHeisenbergalgebragl 1.anyn,thedi erencebetweengl

TherepresentationsofHeisenbergalgebrasinthecontextofgeometricrepresenta-tiontheorywere rstconstructedbyGrojnowski[G]andNakajima[N2](see[N4]forareview).However,itisnotobvioushowtoadaptthisrepresentationtheory (v,w0),obtainingthedesiredcommutationrelationstothenewquivervarietiesL n+1.Thisproblemwillbeconsideredinafuturework.withthegeneratorsofsl

5.ArbitraryLevelRepresentations

5.1.TypeA∞.We rstrecallsomede nitionsfrom[DJKMO2].AMayadiagramisabijectionm:Z→Zsuchthat(m(j))j<0and(m(j))j≥0arebothincreasing.ForeachMayadiagramthereexistsauniqueγ∈Zsuchthatm(j) j=γfor|j| 0.Thisγiscalledthechargeofm.WedenotethesetofMayadiagramsofchargeγbyM[γ].Form∈M[γ]welet

m[r]=(m(j)+r)j∈Z∈M[γ+r].

WecanvisualizeaMayadiagrambyaYoungdiagram.Consideralatticeontherighthalfplanewithlatticepoints{(i,j)∈Z2|i≥0}.Eachedgeonthelatticeisoriented,startingat(i,j)andendingat(i+1,j)or(i,j+1)andisnumberedbytheintegeri+j.ApathonthelatticeisamapefromZtothesetofedgesonthelatticesuchthate(j)hasnumberjandtheendingsiteofe(j)isthestartingsiteofe(j+1).ToeachMayadiagramofchargeγ,weassociatetheuniquepathsatisfyingthefollowingconditions.

(1)Forj 0,e(j)istheedgefrom(0,j)to(0,j+1),

(2)Theedgee(m(j))isvertical(resp.horizontal)ifj<0(resp.j≥0). (v,w0)arepreciselythoseXfTheorem4.3.1.TheirreduciblecomponentsofL Vsuchthatwheref∈Z

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

12IGORB.FRENKELANDALISTAIRSAVAGE

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