Lyashko–Looijenga morphisms and primitive factorizations of Coxeter elements

In the seminal work (Bessis in Ann Math (2) 181(3):809–904, 2015), Bessis gave a geometric interpretation of the noncrossing lattice NC(W) associated to a well-generated complex reflection group W. A chief component of this was the trivialization theorem, a fundamental correspondence between families of chains of NC(W) and the fibers of a finite quasi-homogeneous morphism, the LL map. We consider a variant of the LL map, prescribed by the trivialization theorem, and apply it to the study of finer enumerative and structural properties of NC(W). In particular, we extend work of Bessis and Ripoll and enumerate the so-called “primitive factorizations” of Coxeter elements c. These are the length additive factorizations of the form \(c=w\cdot t_1\cdots t_k\) $c=w·t_1\cdots t_k$ , where w belongs to a given conjugacy class and the \(t_i\) $t_i$ ’s are reflections.