The quaternionic systems of imprimitivity for the reflection groups of rank two
Group Theory
2026-01-27 v1
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).
Cite
@article{arxiv.2601.17075,
title = {The quaternionic systems of imprimitivity for the reflection groups of rank two},
author = {Shayne Waldron},
journal= {arXiv preprint arXiv:2601.17075},
year = {2026}
}