The Loewy length of a tensor product of modules of a dihedral two-group

TL;DR

This paper calculates the Loewy length of tensor products of modules over dihedral 2-groups in characteristic 2, providing new insights into their structure and identifying conditions for projective summands.

Contribution

It introduces a method to compute the Loewy length of tensor products of dihedral 2-group modules, a problem previously unexplored.

Findings

Loewy length formulas for tensor products

Criteria for tensor products to have projective summands

Enhanced understanding of module structure in characteristic 2

Abstract

While the finite-dimensional modules of the dihedral 2-groups over fields of characteristic 2 were classified over 30 years ago, very little is known about the tensor products of such modules. In this article, we compute the Loewy length of the tensor product of two modules of a dihedral two-group in characteristic 2. As an immediate consequence, we determine when such a tensor product has a projective direct summand.