[Paper Review] Divisionally free Restrictions of Reflection Arrangements

Gerhard Roehrle|arXiv (Cornell University)|Oct 1, 2015

Advanced Combinatorial Mathematics4 references3 citations

TL;DR

This paper investigates divisionally free restrictions of reflection arrangements, a class of hyperplane arrangements where Terao's conjecture on the combinatorial nature of freeness holds. It shows that while divisional freeness is preserved under products, it is not closed under localizations—surprisingly, some restrictions of divisionally free arrangements fail to be divisionally free. The key contribution is a complete classification of all divisionally free restrictions of irreducible complex reflection arrangements, identifying precise conditions under which such restrictions remain divisionally free, including exceptional cases for $G_{33}$ and $G_{34}$ with 4-dimensional flats.

ABSTRACT

We study some aspects of divisionally free arrangements which were recently introduced by Abe. Crucially, Terao's conjecture on the combinatorial nature of freeness holds within this class. We show that while it is compatible with products, surprisingly, it is not closed under taking localizations. In addition, we determine all divisionally free restrictions of all reflection arrangements.

Motivation & Objective

• To determine which restrictions of reflection arrangements are divisionally free, extending Abe's classification of divisionally free reflection arrangements.
• To investigate the behavior of divisional freeness under fundamental operations such as products and localizations.
• To resolve the open problem of identifying all divisionally free restrictions of irreducible complex reflection arrangements.
• To clarify the relationship between divisional freeness and hereditary freeness in the context of restricted arrangements.

Proposed method

• The authors use the definition of divisionally free arrangements introduced by Abe, which generalizes inductively free arrangements and ensures Terao's conjecture holds within this class.
• They apply lattice-theoretic techniques and the theory of restriction arrangements to analyze intersections $X \in L(\mathscr{A}(W))$ for complex reflection groups $W$.
• The classification relies on known results on the structure of restrictions of reflection arrangements, particularly from Orlik-Solomon and Shephard-Todd.
• They employ characteristic polynomials and Möbius functions to verify freeness and divisional freeness, especially through the use of the divisibility condition $\chi(\mathscr{A}^X,t) \mid \chi(\mathscr{A},t)$.
• The paper uses product decomposition and lattice isomorphisms to reduce the problem to the irreducible case.
• They verify the results via case-by-case analysis on irreducible reflection groups, especially $G(r,r,\ell)$, $G_{24}, G_{27}, G_{29}, G_{33}, G_{34}$.

Experimental results

Research questions

• RQ1 Which restrictions of reflection arrangements are divisionally free, and what conditions determine this property?
• RQ2 Is the class of divisionally free arrangements closed under taking localizations, and if not, what are the counterexamples?
• RQ3 How does divisional freeness relate to inductive freeness and hereditary freeness in restricted arrangements?
• RQ4 Are there exceptional cases where restrictions of non-divisionally-free groups can still be divisionally free, and if so, under what conditions?

Key findings

• Divisionally free arrangements are preserved under the product construction, extending known closure properties of related classes.
• Surprisingly, divisional freeness is not closed under localizations: there exist divisionally free arrangements whose restrictions are not divisionally free.
• All restrictions of irreducible reflection arrangements are divisionally free if and only if they are inductively free, or fall into specific exceptional cases: $W = G(r,r,\ell)$ with $\dim X = p$ and $1 \leq k \leq p-3$ for $\mathscr{A}^k_p(r)$, or $W = G_{33}$ or $G_{34}$ with $\dim X = 4$.
• The restriction $\mathscr{A}(W)^X$ is divisionally free if and only if it is hereditarily divisionally free, extending a property known for the full reflection arrangement.
• The only exception to divisional freeness among restrictions occurs when $W = G_{34}$ and $\dim X = 5$, where the restriction fails to be divisionally free.
• The classification confirms that for $G(r,r,\ell)$, restrictions $\mathscr{A}^k_p(r)$ are divisionally free precisely when $k \leq p-3$ for $p \geq 4$ and $r,\ell \geq 3$.