Between Ish and Shi
Abstract
We introduce a new family of hyperplane arrangements in dimension that includes both the Shi arrangement and the Ish arrangement. We prove that all the members of a given subfamily have the same number of regions - the connected components of the complement of the union of the hyperplanes - which can be bijectively labeled with the Pak-Stanley labeling. In addition, we show that, in the cases of the Shi and the Ish arrangements, the number of labels with reverse centers of a given length is equal, and conjecture that the same happens with all of the members of the family.
Keywords
Cite
@article{arxiv.1703.02509,
title = {Between Ish and Shi},
author = {Rui Duarte and António Guedes de Oliveira},
journal= {arXiv preprint arXiv:1703.02509},
year = {2018}
}
Comments
In this version we introduced improvements and corrections that have been kindly suggested by the reviewer of Discrete Mathematics, where the article has been accepted for publication