Artin groups and hyperplane arrangements

These are the notes of a mini-course given at the conference “Arrangements in Pyrénées”, held in Pau (France) from 11th to 15th of June, 2012. Definition. Let S be a finite set. A Coxeter matrix on S is a square matrix M = (ms,t)s,t∈S indexed by the elements of S and satisfying: (a) ms,s = 1 for all s ∈ S; (b) ms,t = mt,s ∈ {2, 3, 4, . . . } ∪ {∞} for all s, t ∈ S, s 6= t. A Coxeter matrix is usually represented by its Coxeter graph, Γ = Γ(M). This is a labeled graph defined as follows. (a) S is the set of vertices of Γ. (b) Two vertices s, t ∈ S are connected by an edge if ms,t ≥ 3, and this edge is labeled by ms,t if ms,t ≥ 4. Example. The following matrix is a Coxeter matrix. M = 1 3 2 3 1 4 2 4 1  . It is represented by the Coxeter graph drawn in Figure 1. 4 Figure 1. A Coxeter graph. Definition. Let Γ be a Coxeter graph. The Coxeter system of Γ is defined to be the pair (W,S) = (WΓ, S), where S is the set of vertices of Γ and W is the group presented as follows. WΓ = 〈 S ∣∣∣∣ s = 1 for all s ∈ S (st)s,t = 1 for all s, t ∈ S, s 6= t, ms,t 6=∞ 〉 .