Between Ish and Shi

Abstract

We introduce a new family of hyperplane arrangements in dimension $n\ge 3$ 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.

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

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