The Shi arrangement and the Ish arrangement

TL;DR

This paper explores the relationship between the Shi and Ish arrangements of hyperplanes, revealing a combinatorial symmetry that preserves key properties and extends to generalized arrangements.

Contribution

It introduces a mysterious symmetry between Shi and Ish arrangements, demonstrating that they share key combinatorial invariants and extending results to generalized arrangements.

Findings

Shi and Ish arrangements share the same characteristic polynomial.

The arrangements have the same number of regions, bounded regions, and other combinatorial features.

The symmetry extends to generalized 'deleted' arrangements with arbitrary subgraphs.

Abstract

This paper is about two arrangements of hyperplanes. The first --- the Shi arrangement --- was introduced by Jian-Yi Shi to describe the Kazhdan-Lusztig cells in the affine Weyl group of type $A$. The second --- the Ish arrangement --- was recently defined by the first author who used the two arrangements together to give a new interpretation of the $q,t$-Catalan numbers of Garsia and Haiman. In the present paper we will define a mysterious "combinatorial symmetry" between the two arrangements and show that this symmetry preserves a great deal of information. For example, the Shi and Ish arrangements share the same characteristic polynomial, the same numbers of regions, bounded regions, dominant regions, regions with $c$ "ceilings" and $d$ "degrees of freedom", etc. Moreover, all of these results hold in the greater generality of "deleted" Shi and Ish arrangements corresponding to an arbitrary subgraph of the complete graph. Our proofs are based on nice combinatorial labelings of Shi and Ish regions and a new set partition-valued statistic on these regions.