Algebraic modules and the Auslander–Reiten quiver

Recall that an algebraic module is a KG-module that satisfies a polynomial with integer coefficients, with addition and multiplication given by the 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. For dihedral 2-groups, we also show that there is at most one algebraic module on each component of the (stable) Auslander–Reiten quiver. We include a strong conjecture on the relationship between periodicity and algebraicity.