TY - JOUR
T1 - Affine schur duality
AU - Flicker, Yuval Z.
N1 - Publisher Copyright:
© 2021 Heldermann Verlag.
PY - 2021
Y1 - 2021
N2 - 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 = Cn . Here we extend this duality to the context of the affine Weyl (or symmetric) group Zd oSd and the affine Lie ( or Kac-Moody) algebra eg = Lg⊕Cc, g = sln(C) . Thus we construct a functor F : M 7! M Sd E×d from the category of finite dimensional C[Zd o Sd] -modules M to that of finite dimensional eg -modules W of level 0 (the center Cc of eg acts as zero, thus these are representations of the loop group Lg = L C g, where L = C[t, t-1] , g = sln(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×.
AB - 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 = Cn . Here we extend this duality to the context of the affine Weyl (or symmetric) group Zd oSd and the affine Lie ( or Kac-Moody) algebra eg = Lg⊕Cc, g = sln(C) . Thus we construct a functor F : M 7! M Sd E×d from the category of finite dimensional C[Zd o Sd] -modules M to that of finite dimensional eg -modules W of level 0 (the center Cc of eg acts as zero, thus these are representations of the loop group Lg = L C g, where L = C[t, t-1] , g = sln(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×.
KW - Affine Kac-Moody algebra
KW - Affine Lie algebra
KW - Affine Lie group
KW - Affine Schur duality
KW - Evaluation representations
KW - Finite dimensional representations
KW - Loop algebra
KW - Loop group
UR - http://www.scopus.com/inward/record.url?scp=85105524513&partnerID=8YFLogxK
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AN - SCOPUS:85105524513
SN - 0949-5932
VL - 31
SP - 681
EP - 718
JO - Journal of Lie Theory
JF - Journal of Lie Theory
IS - 3
ER -