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.
University of Minnesota Ph.D. dissertation. September 2017. Major: Mathematics. Advisor: Victor Reiner. 1 computer file (PDF); ix, 95 pages.
Applications of geometric techniques in Coxeter-Catalan combinatorics.
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