Lyashko-Looijenga morphisms and submaximal factorisations of a Coxeter element

TL;DR

This paper explores the geometric and algebraic structures underlying noncrossing partitions of finite reflection groups, providing case-free formulas and new enumeration results for Coxeter element factorizations.

Contribution

It introduces a case-free interpretation of noncrossing partition chains via Lyashko-Looijenga coverings and generalizes existing formulas for reflection groups.

Findings

Case-free interpretation of noncrossing partition chains

Generalized Fuss-Catalan number formulas

New enumeration formulas for Coxeter element factorizations

Abstract

When W is a finite reflection group, the noncrossing partition lattice NCP_W of type W is a rich combinatorial object, extending the notion of noncrossing partitions of an n-gon. A formula (for which the only known proofs are case-by-case) expresses the number of multichains of a given length in NCP_W as a generalised Fuss-Catalan number, depending on the invariant degrees of W. We describe how to understand some specifications of this formula in a case-free way, using an interpretation of the chains of NCP_W as fibers of a Lyashko-Looijenga covering (LL), constructed from the geometry of the discriminant hypersurface of W. We study algebraically the map LL, describing the factorisations of its discriminant and its Jacobian. As byproducts, we generalise a formula stated by K. Saito for real reflection groups, and we deduce new enumeration formulas for certain factorisations of a Coxeter element of W.