Maximal root subsystems of affine reflection systems and duality

TL;DR

This paper classifies maximal root subsystems of affine reflection systems, including non-reduced cases, and characterizes their maximal closed subsystems, with applications to Weyl group subgroups and Saito's EARS.

Contribution

It extends the classification of maximal root subsystems to affine reflection systems and provides explicit criteria for maximal closed subsystems, including nullity 2 cases.

Findings

Classified maximal root subsystems of affine reflection systems.

Determined when a maximal root subsystem is also maximal closed.

Characterized maximal root systems with nullity ≤ 2 using Hermite normal forms.

Abstract

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 result 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 classifies the maximal subgroups of the Weyl group of an affine reflection system that are generated by reflections.