A note on Weyl groups and crystallographic root lattices

TL;DR

This paper investigates criteria for generating Weyl groups using reflections based on root and coroot lattices, and explores special generating sets involving parabolic subgroups, highlighting their simplicity.

Contribution

It introduces a new criterion for when a set of reflections generates a Weyl group, based on root and coroot lattice properties, and examines the structure of special generating sets.

Findings

A criterion for generating Weyl groups from reflections.

Characterization of tame generating sets involving parabolic subgroups.

Insights into the structure of root and coroot lattices.

Abstract

We follow the dual approach to Coxeter systems and show for Weyl groups a criterium which decides whether a set of reflections is generating the group depending on the root and the coroot lattice. Further we study special generating sets involving a parabolic subgroup and show that they are very tame.