Finite quaternionic reflection groups

Abstract

In this article the quaternionic reflection groups are classified. Such a group is defined so as to generalize the notion of reflection groups appearing in [4, 171, i.e., it is a group of linear transformations in a quaternionic vector space of dimension n < cc generated by elements that fix an (n 1)-dimensional subspace pointwise. Moreover, the notion of root system as given in 13, 41 is extended to the quaternionic case. These systems can be used to construct the groups involved and vice versa. Relevant definitions can be found in Section 1. In the following section, the imprimitive groups are determined in a relatively simple manner that resembles the complex case. The primitive groups are classified by means of their complexifications (i.e., their natural isomorphic images in the group of all invertible linear transformations in the 2n-dimensional complex vector space underlying the given quaternionic one). The primitive groups with imprimitive complexifications are dealt with in Section 3. In order to determine the remaining groups, extensive use is made of the work of’Huffman and Wales [12-15, 201 on the classification of primitive unimodular complex linear groups generated by elements that fix a subspace of codimension 2 pointwise. If the complexification of a quaternionic reflection group is primitive, it belongs to this family of groups. Section 4 is about these groups and their root systems. The main results are comprised in Theorems 2.6, 2.9, 3.6, and 4.2.