Affine super Schur duality (Q2693167)

Summary: 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 \(\overline{\mathbb{E}}^{\otimes d}\), \( \overline{\mathbb{E}} = \mathbb{C}^n\). The latter are called polynomial representations homogeneous of degree \(d\). It is based on decomposing \(\overline{\mathbb{E}}^{\otimes d}\) as a \(\mathbb{C} [ S_d ] \times \operatorname{GL} ( n , \mathbb{C} )\)-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 \(\mathcal{F} : M \mapsto M \otimes_{\mathbb{C} [ S_d ]} \mathbb{E}^{\otimes d}\), \( \mathbb{E}\) now being the super vector space \(\mathbb{C}^{m | n}\), \textit{from} the category of finite-dimensional \(\mathbb{C}[S_d \ltimes \mathbb{Z}^d]\)-modules, or representations of the affine Weyl, or symmetric, group \(S_d^a = S_d \ltimes \mathbb{Z}^d\), \textit{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 \(\mathbb{C}^\times\).