Affine Super Schur Duality

Abstract

Schur duality is an equivalence, for $d\leq n$, between the category of finite-dimensional representations over $\mathbb{C}$ of the symmetric group $S\_d$ on $d$ letters, and the category of finite-dimensional representations over $\mathbb{C}$ of $\operatorname{GL}(n,\mathbb{C})$ whose irreducible subquotients are subquotients of $\smash{\overline{\mathbb{E}}^{\otimes d}}$, $\overline{\mathbb{E}}=\mathbb{C}^n$. The latter are called polynomial representations homogeneous of degree $d$. It is based on decomposing $\smash{\overline{\mathbb{E}}^{\otimes d}}$ as a $\mathbb{C}\[S\_d]\times\operatorname{GL}(n,\mathbb{C})$-bimodule. It was used by Schur to conclude the semisimplicity of the category of finite-dimensional complex $\operatorname{GL}(n,\mathbb{C})$-modules from the corresponding result for $S\_d$ that had been obtained by Young. Here we extend this duality to the affine super case by constructing a functor $\mathcal{F}\colon M\mapsto M\otimes\_{\mathbb{C}\[S\_d]}\mathbb{E}^{\otimes d}$, $\mathbb{E}$ now being the super vector space $\mathbb{C}^{m|n}$, from the category of finite-dimensional $\mathbb{C}\[S\_d\ltimes\mathbb{Z}^d]$-modules, or representations of the affine Weyl, or symmetric, group $S\_d^a=S\_d\ltimes\mathbb{Z}^d$, \emph{to} the category of finite-dimensional representations of the universal enveloping algebra of the affine Lie superalgebra $\mathfrak{U}(\widehat{\operatorname{sl}}(m|n))$ that are $\mathbb{E}^{\otimes d}$-compatible, namely the subquotients of whose restriction to $\mathfrak{U}(\operatorname{sl}]\(m|n))$ are constituents of $\mathbb{E}^{\otimes d}$. 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 $\mathbb{E}^{\otimes d}$-compatible representations of the affine superalgebra $\widehat{\operatorname{sl}}(m|n)$ are tensor products of evaluation representations at distinct points of $\mathbb{C}^\times$.