Schur--Weyl Equivalences for Wreath Product Superalgebras

TL;DR

The paper extends Schur--Weyl duality to wreath product superalgebras by constructing a Schur--Weyl functor $F^{A}_{n,d}$ between $S_{d} \wr A$-modules and polynomial representations of degree $d$ for the Lie superalgebra $\mathfrak{gl}_{n}(A)$, proving an equivalence of categories for $n \ge d+1$. It develops a robust framework: defining a polynomial category $\mathcal{P}_{d}(\mathfrak{gl}_{n}(A))$, establishing a faithful and full functor for $n \ge d$, and then proving essential surjectivity when $n \ge d+1$ by reconstructing the $S_{d} \wr A$-action from the $\mathfrak{gl}_{n}(A)$-action. The method bootstraps from classical Schur--Weyl duality, extends to arbitrary associative superalgebras, and recovers several known dualities in special cases, while also yielding structural consequences such as semisimplicity criteria, highest-weight structures, and a parametrization of simple modules via partitions and wreath-product data. The sharp bound $n \ge d+1$ is demonstrated with a concrete counterexample, underscoring the generality and limits of the equivalence and suggesting avenues for further extensions to related superalgebras, characteristic $p$-theory, and quantized or affine analogues.

Abstract

Let $A$ be an associative superalgebra over a field of characteristic zero. Let $n \geq d+1$. The main result of the paper establishes an equivalence of categories between supermodules for the wreath product $ S_{d} \wr A$ and an explicitly defined category of supermodules for the general linear Lie algebra $\mathfrak{gl}_{n}(A)$. We also give an example showing the bound $n \geq d+1$ cannot be improved.

Theorems & Definitions (28)

• Theorem
• Definition 2.4.1
• Remark 2.4.2
• Example 2.4.3
• Remark 2.4.4
• Remark 2.6.1
• Proposition 2.6.2
• Proposition 3.1.1
• proof
• Theorem 3.2.1