Groupoids with root systems in real vector spaces

The study of root systems attached to groupoids that resemble Coxeter groups has seen many recent developments. We take the notion of signed groupoid set, introduced by Dyer to abstract primitive features of such groupoids with root systems, as our general framework.

The main purpose of this thesis is to discuss realizations of root systems of signed groupoid sets in real vector spaces. Inversion sets and the weak and dominance preorders play a major role, and we discuss realizations in relation to these notions. Compressed signed groupoid sets, for which the dominance preorder is a partial order, are particularly important and we give a construction attaching a compressed realized signed groupoid set to an uncompressed one under certain hypotheses.

In some cases, we establish a correspondence between the inversion sets and hy- perplanes that separate the roots of a realization of a compressed signed groupoid set. We then focus on the groupoids defined by Brink and Howlett in their study of nor- malizers of parabolic subgroups of Coxeter groups. The strongest results we obtain hold when the Coxeter group is finite, in which case we give an isomorphism between a realization of the universal covering of the corresponding signed groupoid set and a realized signed groupoid set arising from a simplicial hyperplane arrangement.