Affine Schur Duality

摘要

The Schur duality may be viewed as the study of the commuting actions of the symmetric group Sd and the general linear group GL(n, C) on E. d where E = C-n. Here we extend this duality to the context of the affine Weyl (or symmetric) group Zd o Sd and the affine Lie ( or Kac-Moody) algebra (g) over tilde = L-g circle plus Cc, g = sl(n)(C). Thus we construct a functor F : M M circle times S-d E-circle times d from the category of finite dimensional CZd o Sd -modules M to that of finite dimensional e g -modules W of level 0 (the center Cc of e g acts as zero, thus these are representations of the loop group L-g = L-circle times C g, where L = Ct, t-1, g = sl(n)(C)), the irreducible constituents of whose restriction to g are subrepresentations of E.d. When d < n it is an equivalence of categories, but not for d = n, in contrast to the classical case. As an application we conclude that all irreducible finite dimensional representations of Lg, the irreducible constituents of whose restriction to g are subquotients of E. d, are tensor products of evaluation representations at distinct points of C-x.