Bases of representations of type A affine Lie algebras via q(6)

18IGORB.FRENKELANDALISTAIRSAVAGE

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Figure5.Removingan(n+1)×lsquarefromaMayadiagram

(heren=2andl=4).Noticethattheenumerationofthehori-

zontaledgesdoesnotchangemod(n+1).

Recallthede nitionofRMgiveninSection5.2.ForM,M′∈M[Λ],wesaythatMisann-reductionofM′ifRMisobtainedfromRM′bytheremovalofsetsoftheform{(k+i,l)|0≤i≤n}forsomekandl.

′Proposition6.2.1.SupposeM=(m1,...,ml)isann-reductionofM′=(m′1,...,ml)′′′andm1≤···≤ml≤m1[n+1],m′1≤···≤ml≤m1[n+1].ThenMandMare

liftsofthesamepathandM≥M′.

Proof.RecalltheconstructionintheproofofTheorem5.1.2.Notethatchoosingarbitrarystringsinsteadofthelongeststringateachstepwillnotchangethevalues′′oftherighthandsideof(6.2.2)(foranyk).Thus,letusformM′′=(m′′1,...,ml)∈M[Λ]fromthesamestringscomprisingM′butwhereoneofthem′′icontainstheentiresetofstringsoftheform{(k+i,l)|0≤i≤n}whichisremovedfromRM′toobtainRM.Now,removingthissetofstringsfromM′′simplyamountstoremovingthissetfromm′′i.Butthisjustcutsan(n+1)×lsquareoutoftheMayadiagram′′miandshiftsthepartofthediagrambelowthecutupn+1units.SeeFigure5.

Since µ+n+1= µ,thevaluesoftherighthandsidesof(6.2.2)forM′andM′′arethesame.However,MissimplyobtainedfromM′′byapplyingtheprocedureofTheorem5.1.2tothestringsofM′′andasmentionedabove,thisdoesnotchangethevalueoftherighthandsidesof(6.2.2).ThusMandM′areliftsofthesamepath.

ToshowthatM≥M′,notethatbytheconstructionintheproofofTheo-rem5.1.2,MisuniquelydeterminedbyRM.Now,weobtainRMfromRM′byremovingasetoftheform{(k+i,l)|0≤i≤n}forsomekandl.Thus,ateachstageinourconstructionofM,wechoseastringoflengthlessthanorequaltothestringchosenintheconstructionofM′.ThuswehavethatM≥M′. Proposition6.2.2([DJKMO2]).ForeachΛ-pathηthereexistsauniquehighestliftMofηsuchthatM≥M′foranyliftM′ofη.

Corollary6.2.3.Theset

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

ispreciselythesetofhighestliftsofpathsinP(Λ).

LetMηbethen-reducedelementofM[Λ]correspondingtoη∈P(Λ)andletg

(1)bethea neLiealgebraoftypeAn.De ne

P(Λ)µ={η∈M[Λ]|λη=µ}.

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,QUIVERVARIETIESANDSTATISTICALMECHANICS19In[DJKMO2],theauthorsintroducedabasis{ξη|η∈P(Λ)µ}oftheµweightspaceoftherestricteddualofthehighestweightrepresentationofgofhighestweightΛ

[DJKMO2,Thm5.4].Theweightofξηisλη[DJKMO2,Thm5.7].

Theorem6.2.4.ThemapgXMη→ξηisaweight-preservingvectorspaceiso-morphismbetweenthegeometricpresentationL(w)ofL(Λ)andthepathspacerepresentationof[DJKMO2].

Proof.ThefactthatwehaveavectorspaceisomorphismfollowsfromProposi-tion6.2.2andCorollary6.2.3.Itremainstoshowthatthemapisweight-preserving.Thede nitionofapathagreeswiththede nitionofabasicpathwhenΛ=Λ0andtheweightsarethesameinthiscase.ThuswehavetheresultforΛ=Λ0fromtheprevioussubsection.Thentheresultforarbitrarylevelonerepresentationsfollowseasily.

Now,if

Mη=((Y1,γ1),...,(Yl,γl))

andViisthespacecorrespondingtothestrings(seetheproofofTheorem5.1.2)of(Yi,γi)(thatis,itsdimensionindegreejisequaltothenumberofverticesofthesestringsthatarenumberedj)thentheweightofgMηis

l i=1(Λγi dimVi)

wheredimViisidenti edwithanelementoftherootlatticeasinSection1.But lthisisequaltoi=1ληiwhere(Yi,γi)isaliftofηibythelevel1result.ByProposition5.6of[DJKMO2],thisisληasdesired.

References

[B]V.Baranovsky,Moduliofsheavesonsurfacesandactionoftheoscillatoralgebra.J.Di er-

entialGeometry.55,193–227(2000).

[DJKMO1]E.Date,M.Jimbo,A.Kuniba,T.Miwa,M.Okado,Anewrealizationofthebasic

(1)representationofAn.LettersinMathematicalPhysics.17,51–54(1989).

[DJKMO2]E.Date,M.Jimbo,A.Kuniba,T.Miwa,M.Okado,Paths,MayaDiagramsand (r,C).AdvancedStudiesinPureMathematics.19,149–191(1989).representationsofsl

[G]I.Grojnowski,Instantonsanda nealgebrasI:theHilbertschemeandvertexoperators.

Math.Res.Letters.3,275–291(1996).

[JM]M.JimboandT.Miwa,Solitonsandin nitedimensionalLiealgebras.Publ.RIMS,Kyoto

University.19,943-1001(1983).

[L1]G.Lusztig,Quivers,perversesheaves,andquantizedenvelopingalgebras.JAMS,4,no.2,

365–421(1991)

[L2]G.Lusztig,A nequiversandcanonicalbases.IHES,no.76,111–163(1992).

[N1]H.Nakajima,InstantonsonALEspaces,quivervarieties,andKac-Moodyalgebras.Duke

MathematicalJournal.76,no.2,365–416(1994).

[N2]H.Nakajima,HeisenbergalgebraandHilbertschemesofpointsonprojectivesurfaces.Ann.

ofMath.145,379–388(1997).

[N3]H.Nakajima,QuivervarietiesandKac-Moodyalgebras.DukeMathematicalJournal.91,

no.3,515–560(1998).

[N4]H.Nakajima,LecturesonHilbertschemesofpointsonsurfaces.UniversityLectureSeries,

18,AmericanMathematicalSociety,Providence,RI,1999.

IgorB.Frenkel,

DepartmentofMathematics,YaleUniversity,P.O.Box208283,NEWHAVEN,CT,USA06520-8283;

email:frenkel-igor@yale.edu

AlistairSavage,

DepartmentofMathematics,YaleUniversity,P.O.Box208283,NEWHAVEN,CT,USA06520-8283;

email:alistair.savage@aya.yale.edu