Inductive and divisional posets

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 (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.

arXiv comment28 pages, typos in Acknowledgments corrected

References in corpus (6)

• The freeness of ideal subarrangements of Weyl arrangements
• Divisionally free arrangements of hyperplanes
• Orlik-Solomon-type presentations for the cohomology algebra of toric arrangements
• Combinatorics of certain abelian Lie group arrangements and chromatic quasi-polynomials
• $G$-Tutte polynomials and abelian Lie group arrangements
• Characteristic quasi-polynomials of ideals and signed graphs of classical root systems

Cited by in corpus (1)

• Characteristic quasi-polynomials for deformations of Coxeter arrangements of types A, B, C, and D