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

Reading cut the hyperplanes in a real central arrangement 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 H . Taking only positive expressions in these generators yields a new poset that we call the pure shard monoid . When H is simplicial, its poset of regions is a lattice, so it comes equipped with a pop-stack sorting operator Pop . In this case, we use Pop to define an embedding Crackle of Reading’s shard intersection order into the pure shard monoid. When H is the reflection arrangement of a finite Coxeter group, we also define a poset embedding Snap of the shard intersection order into the positive braid monoid; in this case, our three maps are related by Snap=Crackle· Pop .