Lyashko-Looijenga Morphisms and Geometric Factorizations of a Coxeter Element
A common theme in Combinatorics is an unconditional love for the symmetric group. We like to investigate structural and numerological properties of various objects associated with it. It often happens that such objects and phenomena can be generalized to the other (complex) reflection groups as well.
A problem that goes back to Hurwitz and the 19th century is to enumerate (reduced) factorizations of the long cycle (12..n)? Sn? into factors from prescribed conjugacy classes. In the reflection groups case, it corresponds to enumerating factorizations of a Coxeter element.
Bessis gave a beautiful geometric interpretation of such factorizations by using a variant of the Lyashko-Looijenga (LL) map, a finite morphism coming from Singularity theory. We extend some of Bessis' and Ripoll's work and use the LL map to enumerate the so called "primitive factorizations" of a Coxeter element c. That is, factorizations of the form c=w? t1? tk?, where w? belongs to a prescribed conjugacy class and the ti?'s are reflections.