Pop, Crackle, Snap (and Pow): Some Facets of Shards

TL;DR

This paper introduces a new combinatorial framework involving shards, pure shard monoids, and embeddings that connect geometric arrangements with algebraic structures like braid monoids, revealing new order-theoretic insights.

Contribution

It defines the pure shard monoid and embeddings such as Crackle and Snap, linking shard intersection orders with fundamental groups and braid monoids in Coxeter arrangements.

Findings

Shards correspond to generators of the fundamental group of arrangement complements.

The pure shard monoid provides a new poset structure from positive expressions.

Embeddings relate shard intersection orders to braid monoids via Crackle and Snap maps.

Abstract

Reading cut the hyperplanes in a real central arrangement $\mathcal H$ into pieces called \emph{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 \emph{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 $\mathsf{Snap}=\mathsf{Crackle} \cdot \mathsf{Pop}$.