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

Colin Defant, Nathan Williams

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

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

Abstract

Reading cut the hyperplanes in a real central arrangement 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 . Taking only positive expressions in these generators yields a new poset that we call the \emph{pure shard monoid}. When is simplicial, its poset of regions is a lattice, so it comes equipped with a pop-stack sorting operator . In this case, we use to define an embedding of Reading's shard intersection order into the pure shard monoid. When is the reflection arrangement of a finite Coxeter group, we also define a poset embedding of the shard intersection order into the positive braid monoid; in this case, our three maps are related by .
Paper Structure (40 sections, 39 theorems, 118 equations, 10 figures)

This paper contains 40 sections, 39 theorems, 118 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Introduction
  3. The Salvetti Complex
  4. Shards
  5. Shards and the Salvetti Complex
  6. Pop
  7. Crackle
  8. Snap
  9. Pow
  10. Future Work
  11. The Salvetti Complex
  12. Real Hyperplane Arrangements
  13. The Poset of Regions
  14. The Salvetti Complex
  15. Generators and Relations of the Fundamental Group
  16. ...and 25 more sections

Key Result

Theorem 1.1

Let $\mathcal{H}$ be a central real hyperplane arrangement. Given edges $C' \xrightarrow{e} C$ and $D' \xrightarrow{f} D$ in $\mathrm{Sal}(\mathcal{H})$, we have $\tt_{e} \simeq \tt_f$ if and only if $\Sigma(e) = \Sigma(f)$.

Figures (10)

  • Figure 1: Top: the hyperplane arrangement for the dihedral group $I_2(4)$ has eight regions, and its four hyperplanes are cut into six shards. Bottom: the eight 2-cells of $\mathrm{Sal}(\mathcal{H})$, indicated in blue. One 2-cell is attached for each of the eight homotopies $e_1e_2e_3e_4 \cong e_8e_7e_6e_5, e_2e_3e_4e_5^*\cong e_1^*e_8e_7e_6, \ldots, e_8^*e_1e_2e_3 \cong e_7e_6e_5e_4^*$.
  • Figure 2: The poset $\mathrm{Shard}(\mathcal{H},B)$ for the arrangement of \ref{['fig:rank2']}, which is the reflection arrangement of the dihedral group $I_2(4)$. Gray indicates elements in the image of Reading's embedding of the $I_2(4)$-noncrossing partition lattice (with respect to a certain Coxeter element); see \ref{['sec:noncrossing']}.
  • Figure 3: The interval $[\mathbbm{1},\Delta^2]_{\mathbf{P}^+}$ in the pure shard monoid $\mathbf{P}^+(\mathcal{H},B)$ between the identity element and the full twist $\Delta^2$, where $(\mathcal{H},B)$ is as in \ref{['fig:rank2']}. An edge $p \lessdot p'$ is labeled $i$ when $p'=p \cdot \tt_{\Sigma_i}$. Circled elements are in the image of $\mathsf{Crackle}$. An element is colored gray if it appears as a prefix of a word for $\Delta^2$ using only generators corresponding to noncrossing shards; see \ref{['sec:noncrossing']}.
  • Figure 4: The interval $[\mathbbm{1},\Delta^2]_{\mathbf{B}^+}$ in $\mathrm{Weak}(\mathbf{B}^+(I_2(4)))$, where $I_2(4)$ is the dihedral group of order $8$ with simple reflections $s$ and $t$. The reflection arrangement of $I_2(4)$ is shown in \ref{['fig:rank2']}. Circled elements are in the image of $\mathsf{Snap}$. Gray indicates that the element is the image of a $c$-sortable element under $\mathsf{Snap}$, where $c=st$; see \ref{['sec:noncrossing']}.
  • Figure 5: A redrawing of the interval $[\mathbbm{1},\Delta^2]_{\mathbf{P}^+}$ in the pure shard monoid $\mathbf{P}^+(\mathcal{H},B)$ from \ref{['fig:e_to_delta_P']}, where elements are labeled using the conventions of \ref{['sec:rank2_enumeration']}. Boxed elements are in the image of $\mathsf{Pow}$ (see \ref{['sec:pow']}).
  • ...and 5 more figures

Theorems & Definitions (68)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 2.1: bjorner1990hyperplane
  • Theorem 2.2: salvetti1987topology
  • Theorem 2.3: salvetti1987topology
  • ...and 58 more