On the combinatorics of Tits arrangements

Abstract

In this thesis, we study simplicial arrangements of hyperplanes. Classically, a simplicial arrangement $\mathcal{A}$ in $\mathbb{R}^r$ is a finite set of linear hyperplanes such that every component of $\mathbb{R}^r \setminus \left( \bigcup_{H \in \mathcal{A}} H \right)$ is an open simplicial cone. A short introduction is given in Chapter 1. In Chapter 2, we review the precise definitions and collect some important known results; we also recall the more general concept of a Tits arrangement and the corresponding notions. In Chapter 3, we establish some results for these Tits arrangement. In particular, we give a combinatorial characterization of pairs of Tits arrangements differing by one hyperplane. For this, we introduce weak Dynkin diagrams, which generalize the classical Dynkin diagrams in the crystallographic case. Moreover, we show how one may associate a so called finite reflection groupoid to certain Tits arrangements, generalizing the concept of the Weyl groupoid in the crystallographic case. Furthermore, we classify crystallographic Tits arrangements of rank at least seven containing a chamber whose associated Dynkin diagram is of a certain prescribed type. Chapter 4 contains a classification of affine rank three Tits arrangements whose associated root vectors lie on a projective cubic curve. In Chapter 5, we focus on the classical subject of simplicial line arrangements in $\mathbb{P}^2(\mathbb{R})$. A classification even in this case is still an open problem. However, there exists a catalogue published by Grünbaum listing almost all currently known examples; only four additional arrangements have been discovered by Michael Cuntz. One of the main conjectures in the field is that the current catalogue is complete up to finitely many additions. We prove this conjecture for various special kinds of simplicial arrangements. For instance, we prove that a free simplicial arrangement $\mathcal{A}$ such that $t^\mathcal{A}_i=0$ for $i \geq 6 consists of at most forty lines. We also show that there are only finitely many (combinatorial isomorphism classes of) simplicial line arrangements whose associated root vectors lie on an irreducible projective algebraic curve of fixed (or at least bounded) degree. In the last Chapter 6, we study an interesting phenomenon: given a simplicial arrangement $\mathcal{A}$ in $V:=\mathbb{P}^{r-1}(\mathbb{R})$, one may associate in a natural way certain arrangements in the dual space V^*. It turns out that these dual arrangements are also simplicial in many cases and we give finiteness, classification as well as some experimental results in this setup. For instance, we show how to construct certain sporadic arrangements via reflection groups.