The quaternionic reflection groups with
imprimitive complexifications and the finite
subgroups of GL(2,C)
Shayne Waldron
Abstract:
We give a transparent development of the quaternionic reflection groups which
have imprimitive complexifications, including a description of their reflections and
the corresponding normal reflection subgroups. They are seen to be of rank one
or rank two, and are obtained from minimal complex reflection groups by the
addition of one quaternionic reflection and a complex scalar matrix of finite order.
In the case of the rank two groups, we use our techniques to consider the broader
question of what are the minimal irreducible subgroups of GL(2,C) with a given
collineation group. This leads to a complete classification of the finite irreducible
subgroups of GL(2,C) in terms of the minimal groups, with the groups giving rise
to quaternionic reflection groups located within it.
Keywords:
complex reflection groups, quaternionic reflection groups, imprimitive
complexification, lattice of normal reflection subgroups, reflections, hidden reflections,
collineation groups, roots of reflections, Shephard-Todd classification
Math Review Classification:
Primary 05B30, 15B33, 20C25, 20G20, 51M05, 51M20,
Secondary 15B57, 51E99, 51M15, 65D30.
Length: 60 Pages
Last Updated: 6 November 2026
Availability: