Simple extensions of reflection subgroups of primitive complex reflection groups

TL;DR

This paper classifies reflection subgroups of primitive complex reflection groups and explores their properties, showing that certain generating sets lead to parabolic subgroups, thus extending understanding of subgroup structures.

Contribution

It provides a classification of reflection subgroups and their inclusions in primitive complex reflection groups, highlighting conditions for generating parabolic subgroups.

Findings

Reflection subgroups are classified up to conjugacy.

If a group of rank n is generated by n reflections, all subsets generate parabolic subgroups.

The structure of reflection subgroups is systematically determined.

Abstract

If $G$ is a finite primitive complex reflection group, all reflection subgroups of $G$ and their inclusions are determined up to conjugacy. As a consequence, it is shown that if the rank of $G$ is $n$ and if $G$ can be generated by $n$ reflections, then for every set $R$ of $n$ reflections which generate $G$, every subset of $R$ generates a parabolic subgroup of $G$.