Bijections for the Shi and Ish arrangements

The Shi hyperplane arrangement Shi ( n ) was introduced by Shi to study the Kazhdan–Lusztig cellular structure of the affine symmetric group. The 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 (2013) and Armstrong and Rhoades (2012) 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 graph G on n vertices. The key tools in our bijections are the introduction of an Ish analog of parking functions called rook words and a new instance of the cycle lemma of enumerative combinatorics.