The quaternionic systems of imprimitivity for the reflection groups of rank two
Shayne Waldron
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).
Keywords:
systems of imprimitivity,
irreducible groups of rank two,
imprimitive quaternionic reflection groups,
reflection systems,
binary polyhedral groups,
dicyclic groups,
finite collineation groups.
Math Review Classification:
Primary 15B33, 20F55, 20G20, 51F15;
Secondary 20C25, 51M20.
Length: 28 Pages
Last Updated: 23 January 2026
Availability: