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 $\mathsf{Snap}=\mathsf{Crackle}\cdot \mathsf{Pop}$.