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

Shayne Waldron

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arXiv · 2601.17075

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

Shayne Waldron

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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).

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Shayne Waldron. 2026-01-23. The quaternionic systems of imprimitivity for the reflection groups of rank two. https://arxiv.org/abs/2601.17075

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