Between Shi and Ish

We introduce a new family of hyperplane arrangements in dimension n≥3 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 – whic...

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Bibliographic Details
Main Author: Duarte, Rui (author)
Other Authors: Guedes de Oliveira, António (author)
Format: article
Language:eng
Published: 2018
Subjects:
Online Access:http://hdl.handle.net/10773/21736
Country:Portugal
Oai:oai:ria.ua.pt:10773/21736
Description
Summary:We introduce a new family of hyperplane arrangements in dimension n≥3 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.