Lyashko-Looijenga morphisms and primitive factorizations of the Coxeter element

TL;DR

This paper explores the geometric and combinatorial structure of noncrossing lattices associated with complex reflection groups, extending previous work to enumerate primitive factorizations of Coxeter elements using a variant of the Lyashko-Looijenga morphism.

Contribution

It introduces a modified Lyashko-Looijenga map to study primitive factorizations of Coxeter elements and extends existing enumeration results for these factorizations.

Findings

Extended enumeration of primitive factorizations of Coxeter elements.

Established a new variant of the Lyashko-Looijenga morphism.

Connected geometric interpretation with combinatorial factorizations.

Abstract

In a seminal work, Bessis gave a geometric interpretation of the noncrossing lattice $NC(W)$ associated to a well-generated complex reflection group $W$. 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 the Coxeter element $c$. That is, length additive factorizations of the form $c=w\cdot t_1\cdots t_k$, where $w$ belongs to a given conjugacy class and the $t_i$'s are reflections.