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

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

2009-03-20

arxiv.org/abs/0903.3604v3

Journal of Algebra 323 (2010), pp. 1432-1453

Mathematics

Group Theory

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

Scientific paper

10.1016/j.jalgebra.2009.12.010

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.

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