## Details

Original language | English |
---|---|

Article number | e12829 |

Journal | Journal of the London Mathematical Society |

Volume | 109 |

Issue number | 1 |

Publication status | Published - 20 Dec 2023 |

## Abstract

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.

## ASJC Scopus subject areas

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**Inductive and divisional posets.**/ Pagaria, Roberto; Tran, Tan Nhat; Pismataro , Maddalena et al.

In: Journal of the London Mathematical Society, Vol. 109, No. 1, e12829, 20.12.2023.

Research output: Contribution to journal › Article › Research › peer review

*Journal of the London Mathematical Society*, vol. 109, no. 1, e12829. https://doi.org/10.1112/jlms.12829

*Journal of the London Mathematical Society*,

*109*(1), Article e12829. https://doi.org/10.1112/jlms.12829

}

TY - JOUR

T1 - Inductive and divisional posets

AU - Pagaria, Roberto

AU - Tran, Tan Nhat

AU - Pismataro , Maddalena

AU - Vecchi, Lorenzo

N1 - Publisher Copyright: © 2023 The Authors. Journal of the London Mathematical Society is copyright © London Mathematical Society.

PY - 2023/12/20

Y1 - 2023/12/20

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85174544993&partnerID=8YFLogxK

U2 - 10.1112/jlms.12829

DO - 10.1112/jlms.12829

M3 - Article

VL - 109

JO - Journal of the London Mathematical Society

JF - Journal of the London Mathematical Society

SN - 0024-6107

IS - 1

M1 - e12829

ER -