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)