Algebraic Modules and the Auslander--Reiten Quiver

TL;DR

This paper investigates algebraic modules over group algebras, showing that non-periodic algebraic modules are rare and that high complexity algebraic modules are unique within their Auslander--Reiten quiver component, proposing a conjecture on periodicity.

Contribution

It proves rarity of non-periodic algebraic modules and uniqueness of high complexity algebraic modules in their quiver components, along with a conjecture linking periodicity and algebraicity.

Findings

Non-periodic algebraic modules are very rare.

Algebraic modules with complexity ≥ 3 are unique in their Auslander--Reiten quiver component.

A conjecture relates periodicity to algebraicity.

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 non-periodic algebraic modules are very rare, and that if the complexity of an algebraic module is at least 3, then it is the only algebraic module on its component of the (stable) Auslander--Reiten quiver. We include a strong conjecture on the relationship between periodicity and algebraicity.