Bijections for the Shi and Ish arrangements

TL;DR

This paper establishes bijections between regions of Shi and Ish hyperplane arrangements, revealing deep combinatorial similarities and introducing new tools like rook words and a cycle lemma application.

Contribution

It provides the first explicit bijections between Shi and Ish arrangements, generalizing to subgraph arrangements and introducing rook words as a new combinatorial concept.

Findings

Bijections preserve key statistics between Shi and Ish arrangements.

Generalization to arrangements based on subgraphs of K_n.

Introduction of rook words as Ish analogs of parking functions.

Abstract

The {\sf Shi hyperplane arrangement} Shi(n) was introduced by Shi to study the Kazhdan-Lusztig cellular structure of the affine symmetric group. The {\sf Ish hyperplane arrangement} Ish(n) was introduced by Armstrong in the study of diagonal harmonics. Armstrong and Rhoades discovered a deep combinatorial similarity between the Shi and Ish arrangements. We solve a collection of problems posed by Armstrong and Armstrong-Rhoades by giving bijections between regions of Shi(n) and Ish(n) which preserve certain statistics. Our bijections generalize to the `deleted arrangements' Shi(G) and Ish(G) which depend on a subgraph G of the complete graph K_n on n vertices. The key tools in our bijections are the introduction of an Ish analog of parking functions called {\sf rook words} and a new instance of the cycle lemma of enumerative combinatorics.