Affine Super Schur Duality

Abstract

Schur duality is an equivalence, for $d\le n$, 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 $\mathrm{GL}(n,\mathbb{C})$ whose irreducible subquotients are subquotients of ${\overline{\mathbb{E}}}^{\otimes d}$, $\overline{\mathbb{E}}={\mathbb{C}}^n$. The latter are called polynomial representations homogeneous of degree $d$. It is based on decomposing ${\overline{\mathbb{E}}}^{\otimes d}$ as a $\mathbb{C}[S_d]\times \mathrm{GL}(n,\mathbb{C})$-bimodule. It was used by Schur to conclude the semisimplicity of the category of finite-dimensional complex $\mathrm{GL}(n,\mathbb{C})$-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 $\mathcal{F}:M\mapsto M{\otimes }_{\mathbb{C}[S_d]}{\mathbb{E}}^{\otimes d}$, $\mathbb{E}$ now being the super vector space ${\mathbb{C}}^{m|n}$, from the category of finite-dimensional $\mathbb{C}[S_d⋉{\mathbb{Z}}^d]$-modules, or representations of the affine Weyl, or symmetric, group $S_d^a=S_d⋉{\mathbb{Z}}^d$, \emph{to} the category of finite-dimensional representations of the universal enveloping algebra of the affine Lie superalgebra $\mathfrak{U}(\widehat{\mathrm{sl}}(m|n))$ that are ${\mathbb{E}}^{\otimes d}$-compatible, namely the subquotients of whose restriction to $\mathfrak{U}(\mathrm{sl}](m|n))$ 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 $\widehat{\mathrm{sl}}(m|n)$ are tensor products of evaluation representations at distinct points of ${\mathbb{C}}^{\times }$.