Orbites d'Hurwitz des factorisations primitives d'un élément de Coxeter

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.

25 pages, in French (Abstract in English). Version 3 : last version, published in Journal of Algebra (typos corrected, some minor changes)

References in corpus (3)

• Enumerative properties of generalized associahedra
• Chains in the noncrossing partition lattice
• Non-crossing partitions of type (e,e,r)

Cited by in corpus (7)

• EL-Shellability and Noncrossing Partitions Associated with Well-Generated Complex Reflection Groups
• Symmetric Decompositions and the Strong Sperner Property for Noncrossing Partition Lattices
• Groupes de réflexion, géométrie du discriminant et partitions non-croisées
• Lyashko-Looijenga morphisms and submaximal factorisations of a Coxeter element
• Lyashko-Looijenga morphisms and primitive factorizations of the Coxeter element
• Prefixes of minimal factorisations of a cycle
• Discriminants and Jacobians of virtual reflection groups