On the Vertices of Indecomposable Modules Over Dihedral 2-Groups

TL;DR

This paper determines the vertices of all indecomposable modules over the dihedral group of order 8 in characteristic 2 and proposes a conjectural formula for induced modules from certain subgroups in larger dihedral groups.

Contribution

It provides a complete computation of indecomposable module vertices for D8 and introduces a conjectural formula for induced modules in larger dihedral groups.

Findings

Vertices of all indecomposable modules over D8 are computed.

A conjectural formula for induced modules from dihedral subgroups is proposed.

Partial verification of the formula is presented.

Abstract

Let $k$ be an algebraically closed field of characteristic 2. We compute the vertices of all indecomposable $kD_8$-modules for the dihedral group $D_8$ of order 8. We also give a conjectural formula of the induced module of a string module from $kT_0$ to $kG$ where $G$ is a dihedral group of order $\geq 8$ and where $T_0$ is a dihedral subgroup of index 2 of $G$. Some cases where we verified this formula are given.