Zariski Theorems and Diagrams for Braid Groups

Empirical properties of generating systems for complex reflection groups and their braid groups have been observed by Orlik-Solomon and Broué-Malle-Rouquier, using Shephard-Todd classification. We give a general existence result for presentations of braid groups, which partially explains and generalizes the known empirical properties. Our approach is invariant-theoretic and does not use the classification. The two ingredients are Springer theory of regular elements and a Zariski-like theorem. Introduction Complex reflection groups share many properties with real reflection groups. One of the main difficulties however is that no simple combinatorial description of complex reflection groups (generalizing Coxeter systems) is known. Elementary questions, such as knowing how many reflections are needed to generate the group, do not have satisfactory answers. In [OS], Orlik and Solomon mention the following result (where we have modified the notations to be consistent with the ones used here: d1 ≤ · · · ≤ dr are the degrees, d ∗ 1 ≥ · · · ≥ d ∗ r are the codegrees): (5.5) Theorem. Let W be a finite irreducible unitary reflection group. Then the following conditions are equivalent: (i) di + d ∗ i = dr for i = 1, . . . , r, (ii) ∑r i=1(di + d ∗ i ) = rdr, (iii) di < dr for i = 1, . . . , r, (iv) W may be generated by r reflections, (v) If ζ = exp(2iπ/dr) then there exist generating reflections s1, . . . , sr for W such that the element c = s1 . . . sr has eigenvalues ζd1−1, . . . , ζdr−1 and the element c−1 has eigenvalues ζd ∗ 1+1, . . . , ζd ∗ r+1. However, Orlik and Solomon describe these equivalences as “surprising facts for which we have no further explanation”, the proof relying entirely on case-by-case study, using Shephard-Todd classification. Our The author thanks Lê Dũng Tráng, Fabien Napolitano and Bernard Teissier for useful conversations.