Affine Super Schur Duality To the memory of Goro Shimura

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Abstract

Schur duality is an equivalence, for d ≤ n, between the category of finite-dimensional representations over C of the symmetric group Sd on d letters, and the category of finite-dimensional representations over C of GL(n, C) whose irreducible subquotients are subquotients of E⊗d, E = Cn. The latter are called polynomial representations homogeneous of degree d. It is based on decomposing E⊗d as a C[Sd ] × GL(n, C)-bimodule. It was used by Schur to conclude the semisimplicity of the category of finite-dimensional complex GL(n, C)-modules from the corresponding result for Sd that had been obtained by Young. Here we extend this duality to the affine super case by constructing a functor F: M ↦→ M ⊗C[Sd ] E⊗d, E now being the super vector space Cm|n, from the category of finite-dimensional C[Sd ⋉ ℤd ]-modules, or representations of the affine Weyl, or symmetric, group Sda = Sd ⋉ ℤd, to the category of finite-dimensional representations of the universal enveloping algebra of the affine Lie superalgebraU(ŝl(m|n)) that are E⊗dcompatible, namely the subquotients of whose restriction to U(sl(m|n)) are constituents of E⊗d. Both categories are not semisimple. When d < m+n the functor defines an equivalence of categories. As an application we conclude that the irreducible finite-dimensional E⊗d-compatible representations of the affine superalgebra ŝl(m|n) are tensor products of evaluation representations at distinct points of C×.

Original languageEnglish
Pages (from-to)153-202
Number of pages50
JournalPublications of the Research Institute for Mathematical Sciences
Volume59
Issue number1
DOIs
StatePublished - 2023

Keywords

  • Affine symmetric group
  • affine Lie superalgebra
  • affine Schur duality
  • ŝl(m|n)

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