On the Geometry and Topology of Hyperplane Complements Associated to Complex and Quaternionic Reflection Groups
The Weyl group used in Lie theory can be generalized into reflection groups in more general division algebras; in particular, to complex reflection groups and quaternionic reflection groups. When dealing with these groups, the concept of an associated braid group can come into play. In the real (Coxeter) case, including Weyl groups, this is often described in terms of generators and relations but in the complex case, usually described topologically.
In this paper, I will describe the fruit of efforts to see whether the concept of a braid group (and pure braid group) can be extended from the complex case to the quaternionic case, in particular the category of representations. In Chapter 2, I prove that the cohomology of a hyperplane complement in a quaternionic module can be treated in some ways like the braid group in the complex case, and in particular to create representations corresponding to the representations of the braid group whose pure braid group action is unipotent.
However, in attempting to extend this further, problems arise. In particular, the quaternionification of a complex reflection group is isomorphic to itself through complex conjugation, producing a permutation of its hyperplanes that can be represented through a linear map in the quaternions, but not in the complex numbers, and in Chapter 3 I prove that there is an invariant that can distinguish between the two, as well as others, through the topology of the complex hyperplane complement, extending work of Guerville-Ballé from a single case to the entire infinite group class G(m, m, 3).