Bijections for the Shi and Ish arrangements

The Shi hyperplane arrangementShi(n) was introduced by Shi to study the Kazhdan–Lusztig cellular structure of the affine symmetric group. The Ish hyperplane arrangementIsh(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 GG on nn 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.