Applications of geometric techniques in Coxeter-Catalan combinatorics

Abstract

In the seminal work [Bes 15], Bessis gave a geometric interpretation of the noncrossing lattice NC(W)​​ associated to a well-generated complex reflection group W​​. He used it as a combinatorial recipe to construct the universal covering space of the arrangement complement V⧵ ⋃ H​​, and to show that it is contractible, hence proving the K(π,1)​​ conjecture. Bessis' work however relies on a few properties of NC(W)​​ that are only known via case by case verification. In particular, it depends on the numerological coincidence between the number of chains in NC(W)​​ and the degree of a finite morphism, the LL​​ map. We propose a (partially conjectural) approach that refines Bessis' work and transforms the numerological coincidence into a corollary. Furthermore, we consider a variant of the LL​​ map and apply it to the study of finer enumerative 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⋅ t1⋯ tk​​, where w​​ belongs to a prescribed conjugacy class and the ti​​'s are reflections.