Inductive and divisional posets

TL;DR

This work extends freeness-based concepts from central hyperplane arrangements to the combinatorial realm of posets and abelian (toric) arrangements. By introducing inductive posets and divisional posets, it establishes that divisional posets are factorable and that strictly supersolvable posets are contained in inductive posets, with toric root-system ideals providing substantial inductive behavior. It then shows that inductiveness/divisibility of abelian arrangements is completely determined by their intersection posets, and proves factorability results along with explicit exponent data via induction tables. Localization phenomena are explored, revealing that inductiveness can fail to be preserved under localization, and the theory is specialized to toric arrangements of ideals in root systems of types $A$, $B$, and $C$, yielding new inductive (and sometimes supersolvable) classes with explicit exponents and structure.

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.

Figures (4)

• Figure 1: The weighted partition poset $\Pi_3^w$.
• Figure 2: An inductive poset that is not geometric (left) and an induction table for its inductiveness (right). The elements labelled by $x$ and $y$ do not satisfy the requirement of Definition \ref{['def:GP']}.
• Figure 3: The toric arrangement of a type $B_2$ root system with its poset $\mathcal{P}$ of layers (left) and an induction table for inductiveness (right). The induction table is derived thanks to Theorem \ref{['thm:add-IP']} which deduces that $\mathcal{P}$ is inductive with exponents $\exp(\mathcal{P}) = \{2,2\}$. In addition, $\mathcal{P}$ is supersolvable with the elements of a rank-$1$ M-ideal colored in blue. However, $\mathcal{P}$ is not strictly supersolvable since it has no TM-ideal of rank $1$.
• Figure 4: The poset of layers of the toric arrangement $\mathscr{A}_S$ defined by matrix $S$ in \ref{['eq:matrixS']} and an induction table for its inductiveness.

Theorems & Definitions (102)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Remark 2.4
• Definition 2.5
• Remark 2.6