Pop, Crackle, Snap (and Pow): Some facets of shards

Abstract

Reading cut the hyperplanes in a real central arrangement \mathcal{H} into pieces called shards , which reflect order-theoretic properties of the arrangement. We show that shards naturally index certain generators of the fundamental group of the complement of the complexification of \mathcal{H} . Taking only positive expressions in these generators yields a new poset that we call the pure shard monoid . When \mathcal{H} is simplicial, its poset of regions is a lattice, so it comes equipped with a pop-stack sorting operator \mathsf{Pop} . In this case, we use \mathsf{Pop} to define an embedding \mathsf{Crackle} of Reading’s shard intersection order into the pure shard monoid. When \mathcal{H} is the reflection arrangement of a finite Coxeter group, we also define a poset embedding \mathsf{Snap} of the shard intersection order into the positive braid monoid; in this case, our three maps are related by \mathsf{Snap}=\mathsf{Crackle}\cdot \mathsf{Pop} .