The quaternionic systems of imprimitivity for the reflection groups of rank two

TL;DR

This paper provides a method to compute all systems of imprimitivity for rank two reflection groups over quaternions, unifying various results and revealing the structure of quaternionic systems in these groups.

Contribution

It introduces a straightforward procedure for calculating systems of imprimitivity for rank two reflection groups as quaternionic matrix groups, unifying existing theories.

Findings

Primitive complex reflection groups of rank two have either uncountably many or no quaternionic systems of imprimitivity.

The method applies to all rank two reflection groups, including quaternionic ones.

The work consolidates multiple results in the literature on reflection groups.

Abstract

Given an explicit presentation of a reflection group of rank two (or any rank two group for that matter), we give a simple procedure for calculating all its systems of imprimitivity, when viewed as a matrix group over the quaternions. This is applied to all the reflection groups, in particular the quaternionic reflection groups, thereby unifying a number of results and ideas in the literature. For example, a primitive complex reflection group of rank two has either uncountably many quaternionic systems of imprimitivity (3 cases) or none (16 cases).