6 research outputs found
Structural properties of non-crossing partitions from algebraic and geometric perspectives
The present thesis studies structural properties of non-crossing partitions
associated to finite Coxeter groups from both algebraic and geometric
perspectives. On the one hand, non-crossing partitions are lattices, and on the
other hand, we can view them as simplicial complexes by considering their order
complexes. We make use of these different interpretations and their
interactions in various ways.
The order complexes of non-crossing partitions have a rich geometric
structure, which we investigate in this thesis. In particular, we interpret
them as subcomplexes of spherical buildings. From a more algebraic viewpoint,
we study automorphisms and anti-automorphisms of non-crossing partitions and
their relation to building automorphisms. We also compute the automorphism
groups of non-crossing partitions of type and , provided that
for type . For this, we introduce a new pictorial representation for type
.
In type we study the structural properties of the order complex of the
non-crossing partitions in more detail. In particular, we investigate the
interaction of chamber distances and convex hulls in the non-crossing partition
complex and the ambient spherical building. These questions are connected to
the curvature conjecture of Brady and McCammond.Comment: 181 pages, doctoral dissertation, Karlsruher Instituts f\"ur
Technologie (2018