Complexity for Modules Over the Classical Lie Superalgebra gl(m|n)

TL;DR

This paper computes the complexity of simple and Kac modules over the classical Lie superalgebra gl(m|n), revealing a relationship between complexity and atypicality of the modules' blocks.

Contribution

It provides explicit calculations of module complexities for gl(m|n), linking them to the modules' atypicality, extending previous bounds on growth rates.

Findings

Complexity of simple modules is computed.

Complexity of Kac modules is determined.

Complexity relates to the atypicality of the module's block.

Abstract

Let $\mathfrak{g}=\mathfrak{g}_{\bar{0}}\oplus \mathfrak{g}_{\bar{1}}$ be a classical Lie superalgebra and $\mathcal{F}$ be the category of finite dimensional $\mathfrak{g}$-supermodules which are completely reducible over the reductive Lie algebra $\mathfrak{g}_{\bar{0}}$. In an earlier paper the authors demonstrated that for any module $M$ in $\mathcal{F}$ the rate of growth of the minimal projective resolution (i.e., the complexity of $M$) is bounded by the dimension of $\mathfrak{g}_{\bar{1}}$. In this paper we compute the complexity of the simple modules and the Kac modules for the Lie superalgebra $\mathfrak{gl}(m|n)$. In both cases we show that the complexity is related to the atypicality of the block containing the module.