# Inductive Freeness of Ziegler's Canonical Multiderivations for Reflection Arrangements

## TL;DR

This paper investigates the inductive freeness of Ziegler's canonical multiarrangements derived from reflection arrangements, establishing a criterion linking the inductive freeness of the restriction to that of the entire arrangement.

## Contribution

It proves that Ziegler's canonical multiarrangements are inductively free if and only if their restrictions are inductively free, for reflection arrangements.

## Key findings

- Ziegler's canonical multiarrangements are inductively free iff their restrictions are
- The result applies specifically to reflection arrangements of complex reflection groups
- Provides a criterion for inductive freeness in the context of hyperplane arrangements

## Abstract

Let $A$ be a free hyperplane arrangement. In 1989, Ziegler showed that the restriction $A''$ of $A$ to any hyperplane endowed with the natural multiplicity is then a free multiarrangement. We initiate a study of the stronger freeness property of inductive freeness for these canonical free multiarrangements and investigate them for the underlying class of reflection arrangements. More precisely, let $A = A(W)$ be the reflection arrangement of a complex reflection group $W$. By work of Terao, each such reflection arrangement is free. Thus so is Ziegler's canonical multiplicity on the restriction $A''$ of $A$ to a hyperplane. We show that the latter is inductively free as a multiarrangement if and only if $A''$ itself is inductively free.