On the conjecture of Athanasiadis related to freeness of a family of hyparplane arrangements

TL;DR

This paper proves a conjecture by Athanasiadis that characterizes when certain hyperplane arrangements, situated between Coxeter and Catalan arrangements of type A, are free, completing the proof of the conjecture.

Contribution

It provides a complete proof of Athanasiadis's conjecture on the freeness of hyperplane arrangements between Coxeter and Catalan arrangements of type A.

Findings

Confirmed the conjecture for arrangements between Coxeter and Catalan types

Established a characterization of freeness for these arrangements

Completed the proof of Athanasiadis's conjecture

Abstract

We prove a characterization of freeness, conjectured by Athanasiadis, for the family of hyperplane arrangements which lie between the Coxeter and the Catalan arrangement of type $A_\ell$. One direction was already proved in [2]. Here we prove the other direction