Freeness for restriction arrangements of the extended Shi and Catalan arrangements

TL;DR

This paper investigates the freeness properties of cones of extended Shi and Catalan arrangements of type A, establishing hereditary freeness in some cases and characterizing freeness via graphical conditions.

Contribution

It proves hereditary freeness for the cone of the extended Catalan arrangement and characterizes freeness for the extended Shi arrangement using digraph-based conditions.

Findings

Cone of extended Catalan arrangement of type A is hereditarily free.

Freeness of the cone of extended Shi arrangement is determined by dimension.

Graphical conditions characterize freeness of these arrangements.

Abstract

The extended Shi and Catalan arrangements are well investigated arrangements. In this paper, we prove that the cone of the extended Catalan arrangement of type A is always hereditarily free, while we determine the dimension in which the cone of the extended Shi arrangement of type A is hereditarily free. For this purpose, using digraphs, we define a class of arrangements which is closed under restriction, and which contains the extended Shi and Catalan arrangements. We also characterize the freeness for the cone of this arrangement by graphical conditions.