Reflection subgroups of finite and affine Weyl groups

TL;DR

This paper classifies reflection subgroups of finite and affine Weyl groups using root systems and Dynkin diagrams, providing new proofs and detailed classifications with combinatorial and geometric insights.

Contribution

It offers a concise, case-free proof of the classification of reflection subgroups and extends the classification to affine Weyl groups with detailed combinatorial and geometric analysis.

Findings

A short, case-free proof of the classification of reflection subgroups.

Finer classifications of affine reflection subgroups are provided.

Connections between root subsets, alcove geometry, and the Tits cone are established.

Abstract

We discuss the classification of reflection subgroups of finite and affine Weyl groups from the point of view of their root systems. A short case free proof is given of the well known classification of the isomorphism classes of reflection subgroups using completed Dynkin diagrams, for which there seems to be no convenient source in the literature. This is used as a basis for treating the affine case, where finer classifications of reflection subgroups are given, and combinatorial aspects of root systems are shown to appear. Various parameter sets for certain types of subsets of roots are interpreted in terms of alcove geometry and the Tits cone, and combinatorial identities are derived.