Arrangements of ideal type are inductively free

TL;DR

This paper proves that all arrangements of ideal type derived from positive roots in a root system are inductively free, confirming a longstanding conjecture and extending previous results on their freeness.

Contribution

It confirms the conjecture that all arrangements of ideal type are inductively free, expanding the understanding of their algebraic and combinatorial structure.

Findings

All arrangements of ideal type are inductively free.

Confirms a conjecture by R"ohrle.

Extends previous results on freeness of arrangements.

Abstract

Extending earlier work by Sommers and Tymoczko, in 2016 Abe, Barakat, Cuntz, Hoge, and Terao established that each arrangement of ideal type $\mathcal{A}_\mathcal{I}$ stemming from an ideal $\mathcal{I}$ in the set of positive roots of a reduced root system is free. Recently, R\"ohrle showed that a large class of the $\mathcal{A}_\mathcal{I}$ satisfy the stronger property of inductive freeness and conjectured that this property holds for all $\mathcal{A}_\mathcal{I}$. In this article, we confirm this conjecture.