Inductive and Recursive Freeness of Localizations of Multiarrangements

Hoge, Torsten; Röhrle, Gerhard; Schauenburg, Anne

doi:10.1007/978-3-319-70566-8_16

Cited by 4 publications

(2 citation statements)

References 18 publications

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“…Finally, we give a discussion on the localization at a layer of an abelian arrangement (Section 6). It is shown that inductive freeness of a hyperplane arrangement is preserved under taking localization [14]. We show that it is not the case for an arbitrary abelian arrangement by providing an example of an inductive toric arrangement with a noninductive localization.…”

Section: Introductionmentioning

confidence: 94%

Inductive and divisional posets

Pagaria,

Pismataro,

Tran

et al. 2023

Journal of London Math Soc

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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 , or with respect to the root lattice is inductive.

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Section: Introductionmentioning

confidence: 94%

Inductive and divisional posets

Pagaria,

Pismataro,

Tran

et al. 2023

Journal of London Math Soc

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“…This is followed by a discussion of inductive freeness for multiarrangements in Section 2.7. Here we recall results from [HRS17] which show the compatibility of this notion with products and localization for multiarrangements that are used in the sequel.…”

Section: Introductionmentioning

confidence: 95%

Inductive freeness of Ziegler's canonical multiderivations for reflection arrangements

Hoge

Röhrle

2018

Journal of Algebra

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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.

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Inductive Freeness of Ziegler’s Canonical Multiderivations

Hoge,

Röhrle

2024

Discrete Comput Geom

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Let $${{\mathscr {A}}}$$ A be a free hyperplane arrangement. In 1989, Ziegler showed that the restriction $${{\mathscr {A}}}''$$ A ′ ′ of $${{\mathscr {A}}}$$ A to any hyperplane endowed with the natural multiplicity $$\kappa $$ κ is then a free multiarrangement $$({{\mathscr {A}}}'',\kappa )$$ ( A ′ ′ , κ ) . The aim of this paper is to prove an analogue of Ziegler’s theorem for the stronger notion of inductive freeness: if $${{\mathscr {A}}}$$ A is inductively free, then so is the multiarrangement $$({{\mathscr {A}}}'',\kappa )$$ ( A ′ ′ , κ ) . In a related result we derive that if a deletion $${{\mathscr {A}}}'$$ A ′ of $${{\mathscr {A}}}$$ A is free and the corresponding restriction $${{\mathscr {A}}}''$$ A ′ ′ is inductively free, then so is $$({{\mathscr {A}}}'',\kappa )$$ ( A ′ ′ , κ ) —irrespective of the freeness of $${{\mathscr {A}}}$$ A . In addition, we show counterparts of the latter kind for additive and recursive freeness.

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Inductive and Recursive Freeness of Localizations of Multiarrangements

Cited by 4 publications

References 18 publications

Inductive and divisional posets

Inductive and divisional posets

Inductive freeness of Ziegler's canonical multiderivations for reflection arrangements

Inductive Freeness of Ziegler’s Canonical Multiderivations

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