Groupes de r\'eflexion, g\'eom\'etrie du discriminant et partitions non-crois\'ees

TL;DR

This work explores the geometric and algebraic structures underlying noncrossing partitions in complex reflection groups, providing a uniform, case-free explanation for combinatorial formulas related to invariant degrees and factorisations.

Contribution

It introduces a geometric approach using the Lyashko-Looijenga covering to explain combinatorial formulas in noncrossing partitions uniformly across all reflection groups.

Findings

Established a case-free proof for formulas counting submaximal factorizations.

Linked fibers of the Lyashko-Looijenga covering to block factorizations of Coxeter elements.

Derived formulas involving degrees of discriminants and Jacobians for complex reflection groups.

Abstract

Reflection groups, geometry of the discriminant and noncrossing partitions. When W is a well-generated complex reflection group, the noncrossing partition lattice NCP_W of type W is a very rich combinatorial object, extending the notion of noncrossing partitions of an n-gon. This structure appears in several algebraic setups (dual braid monoid, cluster algebras...). Many combinatorial properties of NCP_W are proved case-by-case, using the classification of reflection groups. It is the case for Chapoton's formula, expressing the number of multichains of a given length in the lattice NCP_W, in terms of the invariant degrees of W. This thesis work is motivated by the search for a geometric explanation of this formula, which could lead to a uniform understanding of the connections between the combinatorics of NCP_W and the invariant theory of W. The starting point is to use the Lyashko-Looijenga covering (LL), based on the geometry of the discriminant of W. In the first chapter, some topological constructions of Bessis are refined, allowing to relate the fibers of LL with block factorisations of a Coxeter element. Then we prove a transitivity property for the Hurwitz action of the braid group B_n on certain factorisations. Chapter 2 is devoted to certain finite polynomial extensions, and to properties about their Jacobians and discriminants. In Chapter 3, these results are applied to the extension defined by the covering LL. We deduce --- with a case-free proof --- formulas for the number of submaximal factorisations of a Coxeter element in W, in terms of the homogeneous degrees of the irreducible components of the discriminant and Jacobian for LL.