Bases of representations of type A affine Lie algebras via q(12)

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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Figure3.TheMayadiagramcorresponding

(m(j))j≥0=( 3, 2,0,3,5,6,7,...),(m(j))j<0

(..., 6, 5, 4,, 1,1,2,4).to=

Notethattheseconditionsimplythatforj 0,e(j)istheedgefrom(j γ,γ)to(j γ+1,γ).SeeFigure3.

Suchapathdividestherighthalfplaneintotwocomponents.Theupperhalfisanin niteYoungdiagramYwhichconsistsofaquadrantanda( nite)YoungdiagramYattachedalongahorizontallineatheightγ.ThusthesetofMayadiagramsareinone-to-onecorrespondencewiththesetofpairs(Y,γ)whereY∈Yandγ∈Z.

Lemma5.1.1([DJKMO2]).Letm∈M[γ],m′∈M[γ′],andletY,Y′bethecor-respondingin niteYoungdiagrams.Thenthefollowingconditionsareequivalent.

(1)m(j)≤m′(j)forj≥0,

(2)γ≤γ′andm(j γ)≥m′(j γ′)forj<γ,

(3)Y Y′.

WeputapartialorderingonthesetofMayadiagramsbylettingm≤m′iftheconditionsinLemma5.1.1hold.

LetΛ=Λγ1+...Λγlwhereγ1≤···≤γlandtheΛiarefundamentalweightsofg.Letw∈(Z≥0)Z(thatis,wisfunctionfromZtoZ≥0)bethevectorwithithcomponentequaltothenumberofγjequaltoi.Let

M[Λ]=M[γ1]×···×M[γl].

ForY=[l1,...,ls]∈Y,letAγ

Ybetheset{(γ+1 i,γ+li i)|1≤i≤s}.Forγi ∞betheM=((Y1,γ1),...,(Yl,γl))∈M[Λ],letAM=∪li=1AYiandletfM∈Z

functionsuchthatf(k′,k)isequaltothenumberoftimes(k′,k)appearsinthesetAM.

Theorem5.1.2.TheirreduciblecomponentsofL(v,w)arepreciselythoseXfwheref=fMforsomeM∈M[Λ].DenotethecomponentXfMbyXM.ThenM XMisanatural1-1correspondencebetweentheset

{(m1,...,ml)∈M[Λ]|m1≤···≤ml}

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