Inductive freeness of Ziegler's canonical multiderivations for reflection 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.

Inductive Freeness of Ziegler's Canonical Multiderivations

“…In view of Example 1.15, with the aid of Theorems 1.3 and 1.6, it is feasible to obtain a classification of all instances when the Ziegler restriction is inductively free where the underlying simple arrangement is a restriction of a complex reflection arrangement A (W ); this is carried out in [HRW22].…”