Bases of representations of type A affine Lie algebras via q(16)

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

16IGORB.FRENKELANDALISTAIRSAVAGE

Proof.LetY=[l1,...,ls].ThengXY∈L(v,w0)where

v=dim

=s i=1V(1 i,li i)α¯l.sli i i=1l=1 i

RecallthattheweightofthespaceL(v,w0)is

(u0,...,un)=w0 Cv

andthusHkgXY=ukgXYwith

uk=Λ0(αk)

=δ(k,0)

=δ(k,0) lsi i αk,α¯l (2δ(k,l) δ(k,l 1) δ(k,l+1))i=1l=1 ii=1l=1 is i=1

s i=1sli i (δ(k,1 i) δ(k, i)+δ(k,li i) δ(k,li i+1))(δ(k,li i+1) δ(k,li i))=δ(k, s)+

= k(Y).

Proposition6.1.2.Onehasd(gXY)= ω(Y)gXY.

Proof.We rstcomputethelefthandside.Itisobviousthat

d(gXY)= v0gXY

whereXY L(v,w0).Considertherepresentation(V(k′,k′+l 1),x(k′,k′+l 1))wherel=(n+1)a+bwith0≤b≤n.Then

v0=dimV(k′,k′+l 1)0 1if=a+.k′ 1+b≤n

Thus,forY=[l1,...,ls]∈Ynwhereli=(n+1)ai+biwith0≤bi≤n, s 1ifai+v0=

i=1

1 i,bi i).

Herethereareairepetitionsof0,1,...,nif

i 1≥bi.

The rstlipositionsofthebasicpathcorrespondingtoYiaresimplyobtainedfromtheabovebyloweringalltheentriesby1(interpreting-1asn).Theentries

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