On Morita theory for self-dual modules

TL;DR

This paper explores how Morita equivalences between blocks of group algebras preserve the property of modules being self-dual and, in certain cases, preserve the geometric type of simple modules, especially over fields of odd characteristic.

Contribution

It establishes that self-dual Morita bimodules induce correspondences that preserve self-duality and, when symmetric, also preserve the geometric type of simple modules in odd characteristic.

Findings

Self-dual Morita bimodules preserve self-duality of modules.

Symmetric bilinear forms on bimodules preserve geometric type in odd characteristic.

Preservation of properties extends to projective modules in characteristic 2.

Abstract

Let $G$ be a finite group and let $k$ be a field of characteristic $p$. It is known that a $kG$-module $V$ carries a non-degenerate $G$-invariant bilinear form $b$ if and only if $V$ is self-dual. We show that whenever a Morita bimodule $M$ which induces an equivalence between two blocks $B(kG)$ and $B(kH)$ of group algebras $kG$ and $kH$ is self-dual then the correspondence preserves self-duality. Even more, if the bilinear form on $M$ is symmetric then for $p$ odd the correspondence preserves the geometric type of simple modules. In characteristic 2 this holds also true for projective modules.