2022/09/12 by Defant, Colin, Williams, Nathan
#05E16 #06F15 #20F34 #20F36 #52C35 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2209.05392
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 have a natural interpretation as 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 Pop. In this case, we use Pop to define an embedding 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 Snap of the shard intersection order into the positive braid monoid; in this case, our three maps are related by Snap=Crackle ⋅ Pop.