Inductive Freeness of Ziegler's Canonical Multiderivations for Reflection Arrangements

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.Comment: 23 pages; v2 minor changes; final version, to appear in J. Algebr

Inductive Freeness of Ziegler's Canonical Multiderivations

Let $\mathcal A$ be a free hyperplane arrangement. In 1989, Ziegler showed that the restriction $\mathcal A"$ of $\mathcal A$ to any hyperplane endowed with the natural multiplicity $\kappa$ is then a free multiarrangement $(\mathcal A",\kappa)$. The aim of this paper is to prove an analogue of Ziegler's theorem for the stronger notion of inductive freeness: if $\mathcal A$ is inductively free, then so is the multiarrangement $(\mathcal A",\kappa)$. In a related result we derive that if a deletion $\mathcal A'$ of $\mathcal A$ is free and the corresponding restriction $\mathcal A"$ is inductively free, then so is $(\mathcal A",\kappa)$ -- irrespective of the freeness of $\mathcal A$. In addition, we show counterparts of the latter kind for additive and recursive freeness.Comment: 20 pages. arXiv admin note: text overlap with arXiv:1705.0276

Inductive and divisional posets

We call a poset factorable if its characteristic polynomial has all positive integer roots. Inspired by inductive and divisional freeness of a central hyperplane arrangement, we introduce and study the notion of inductive posets and their superclass of divisional posets. It then motivates us to define the so-called inductive and divisional abelian (Lie group) arrangements, whose posets of layers serve as the main examples of our posets. Our first main result is that every divisional poset is factorable. Our second main result shows that the class of inductive posets contains strictly supersolvable posets, the notion recently introduced due to Bibby and Delucchi (2022). This result can be regarded as an extension of a classical result due to Jambu and Terao (1984), which asserts that every supersolvable hyperplane arrangement is inductively free. Our third main result is an application to toric arrangements, which states that the toric arrangement defined by an arbitrary ideal of a root system of type $A$, $B$ or $C$ with respect to the root lattice is inductive.Comment: 28 pages, typos in Acknowledgments correcte

Inductive and divisional posets

We call a poset factorable if its characteristic polynomial has all positive integer roots. Inspired by inductive and divisional freeness of a central hyperplane arrangement, we introduce and study the notion of inductive posets and their superclass of divisional posets. It then motivates us to define the so-called inductive and divisional abelian (Lie group) arrangements, whose posets of layers serve as the main examples of our posets. Our first main result is that every divisional poset is factorable. Our second main result shows that the class of inductive posets contains strictly supersolvable posets, the notion recently introduced due to Bibby and Delucchi (2022). This result can be regarded as an extension of a classical result due to Jambu and Terao (Adv. in Math. 52 (1984) 248–258), which asserts that every supersolvable hyperplane arrangement is inductively free. Our third main result is an application to toric arrangements, which states that the toric arrangement defined by an arbitrary ideal of a root system of type (Formula presented.), (Formula presented.) or (Formula presented.) with respect to the root lattice is inductive