Séminaire Lotharingien de Combinatoire, 89B.3 (2023), 12 pp.
Colin Defant and Nathan Williams
Pop, Crackle, Snap (and Pow): Some Facets of Shards
Abstract.
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 have a natural interpretation as 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.
Received: November 15, 2022.
Accepted: February 20, 2023.
Final version: April 1, 2023.
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