Pop, Crackle, Snap (and Pow): Some Facets of Shards
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866909070197260288 |
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| author | Defant, Colin Williams, Nathan |
| author_facet | Defant, Colin Williams, Nathan |
| contents | 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 $\mathsf{Snap}=\mathsf{Crackle} \cdot \mathsf{Pop}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2209_05392 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Pop, Crackle, Snap (and Pow): Some Facets of Shards Defant, Colin Williams, Nathan Combinatorics 05E16, 20F34, 20F36, 06F15, 52C35 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 $\mathsf{Snap}=\mathsf{Crackle} \cdot \mathsf{Pop}$. |
| title | Pop, Crackle, Snap (and Pow): Some Facets of Shards |
| topic | Combinatorics 05E16, 20F34, 20F36, 06F15, 52C35 |
| url | https://arxiv.org/abs/2209.05392 |