Orbites d'Hurwitz des factorisations primitives d'un \'el\'ement de Coxeter

TL;DR

This paper investigates the Hurwitz action on primitive factorisations of Coxeter elements in complex reflection groups, aiming to provide a geometric proof of Chapoton's formula through advanced algebraic and geometric methods.

Contribution

It establishes a transitivity property of the Hurwitz action on primitive factorisations, extending known results from reflection decompositions to primitive cases.

Findings

Proves transitivity of Hurwitz action on primitive factorisations.

Connects geometric properties of discriminants to combinatorial formulas.

Provides a new geometric proof of Chapoton's formula.

Abstract

We study the Hurwitz action of the classical braid group on factorisations of a Coxeter element c in a well-generated complex reflection group W. It is well-known that the Hurwitz action is transitive on the set of reduced decompositions of c in reflections. Our main result is a similar property for the primitive factorisations of c, i.e. factorisations with only one factor which is not a reflection. The motivation is the search for a geometric proof of Chapoton's formula for the number of chains of given length in the non-crossing partitions lattice NCP_W. Our proof uses the properties of the Lyashko-Looijenga covering and the geometry of the discriminant of W.