Maximal Root Subsystems of Affine Reflection Systems and Duality

Any maximal root subsystem of a finite crystallographic reduced root system is either a closed root subsystem or its dual is a closed root subsystem in the dual root system. In this article, we classify the maximal root subsystems of an affine reflection system (reduced and non-reduced) and prove that this closed or dual-closed property holds in much more generality for reduced affine reflection systems. Moreover, we explicitly determine when a maximal root subsystem is a maximal closed root subsystem. Using our classification, at the end, we characterize the maximal root systems of affine reflection systems with nullity less than or equal to 2 using Hermite normal forms; especially for Saito’s EARS of nullity 2. This in turn, in the reduced case, classifies the maximal subgroups of the Weyl group of an affine reflection system that are generated by reflections.