Affine Super Schur Duality
Yuval Z. Flicker
Ariel University, Israel; The Ohio State University, Columbus, USA
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Abstract
Schur duality is an equivalence, for , between the category of finite-dimensional representations over of the symmetric group on letters, and the category of finite-dimensional representations over of whose irreducible subquotients are subquotients of , . The latter are called polynomial representations homogeneous of degree . It is based on decomposing as a -bimodule. It was used by Schur to conclude the semisimplicity of the category of finite-dimensional complex -modules from the corresponding result for that had been obtained by Young. Here we extend this duality to the affine super case by constructing a functor , now being the super vector space , from the category of finite-dimensional -modules, or representations of the affine Weyl, or symmetric, group , \emph{to} the category of finite-dimensional representations of the universal enveloping algebra of the affine Lie superalgebra that are -compatible, namely the subquotients of whose restriction to are constituents of . Both categories are not semisimple. When the functor defines an equivalence of categories. As an application we conclude that the irreducible finite-dimensional -compatible representations of the affine superalgebra are tensor products of evaluation representations at distinct points of .
Cite this article
Yuval Z. Flicker, Affine Super Schur Duality. Publ. Res. Inst. Math. Sci. 59 (2023), no. 1, pp. 153–202
DOI 10.4171/PRIMS/59-1-5