Relation spaces of hyperplane arrangements and modules defined by graphs of fiber zonotopes

TL;DR

This paper investigates the algebraic and combinatorial properties of hyperplane arrangements, especially Coxeter arrangements, by analyzing relation complexes and modules derived from fiber zonotopes, providing new bounds and proofs.

Contribution

It introduces new results on the exactness of relation complexes for hyperplane arrangements and establishes bounds on the Castelnuovo-Mumford regularity of related modules.

Findings

Proves n-formality of hyperplane arrangements of reflection groups.

Provides bounds on the Castelnuovo-Mumford regularity of modules.

Simplifies the proof of exactness for relation complexes.

Abstract

We study the exactness of certain combinatorially defined complexes which generalize the Orlik-Solomon algebra of a geometric lattice. The main results pertain to complex reflection arrangements and their restrictions. In particular, we consider the corresponding relation complexes and give a simple proof of the $n$-formality of these hyperplane arrangements. As an application, we are able to bound the Castelnouvo-Mumford regularity of certain modules over polynomial rings associated to Coxeter arrangements (real reflection arrangements) and their restrictions. The modules in question are defined using the relation complex of the Coxeter arrangement and fiber polytopes of the dual Coxeter zonotope. They generalize the algebra of piecewise polynomial functions on the original arrangement.