Structural properties of non-crossing partitions from algebraic and geometric perspectives

TL;DR

This thesis explores the algebraic and geometric structures of non-crossing partitions related to finite Coxeter groups, revealing their lattice and simplicial complex properties, automorphism groups, and connections to spherical buildings and curvature conjectures.

Contribution

It introduces a new pictorial representation for type D non-crossing partitions and analyzes their automorphisms, automorphism groups, and geometric structures from multiple perspectives.

Findings

Automorphism groups of type B and D non-crossing partitions computed.

New pictorial representation for type D introduced.

Connections established between non-crossing partitions and spherical buildings.

Abstract

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 $B$ and $D$, provided that $n \neq 4$ for type $D$. For this, we introduce a new pictorial representation for type $D$. In type $A$ 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.

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