Bases of representations of type A affine Lie algebras via q(19)

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.

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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

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