The freeness of Ish arrangements

TL;DR

This paper proves that the Ish arrangement is supersolvable and therefore free, and provides a criterion for when a deleted Ish arrangement remains free, contributing to the understanding of its algebraic structure.

Contribution

It establishes the supersolvability and freeness of the Ish arrangement and characterizes the conditions for the freeness of its deletions, advancing the theory of hyperplane arrangements.

Findings

Ish arrangement is supersolvable and free.

Necessary and sufficient condition for free deleted Ish arrangements.

Connections to $q,t$-Catalan numbers and combinatorial structures.

Abstract

The Ish arrangement was introduced by Armstrong to give a new interpretation of the $q,t$-Catalan numbers of Garsia and Haiman. Armstrong and Rhoades showed that there are some striking similarities between the Shi arrangement and the Ish arrangement and posed some problems. One of them is whether the Ish arrangement is a free arrangement or not. In this paper, we verify that the Ish arrangement is supersolvable and hence free. Moreover, we give a necessary and sufficient condition for the deleted Ish arrangement to be free.