On the Tensor Products of Modules for Dihedral 2-Groups

TL;DR

This paper investigates algebraic modules within the Auslander-Reiten quiver of dihedral 2-groups, establishing that non-periodic components contain at most one algebraic module, thus advancing understanding of module structures.

Contribution

It proves the uniqueness of algebraic modules in non-periodic components of the Auslander-Reiten quiver for dihedral 2-groups.

Findings

At most one algebraic module per non-periodic component

Clarifies structure of algebraic modules in dihedral 2-groups

Enhances classification of modules in representation theory

Abstract

Recall that an algebraic module is a KG-module that satisfies a polynomial with integer coefficients, with addition and multiplication given by direct sum and tensor product. In this article we prove that if L is a component of the (stable) Auslander-Reiten quiver for a dihedral 2-group consisting of non-periodic modules, then there is at most one algebraic module on L.