Characteristic Polynomials of Deformations of Coxeter Arrangements via levels of regions

TL;DR

This paper introduces a new formula for the characteristic polynomials of deformed Coxeter arrangements using levels of regions, simplifying proofs and extending results to type B arrangements.

Contribution

It provides a novel, simplified formula for characteristic polynomials of deformed Coxeter arrangements, generalizing previous results and applying the concept of levels of regions.

Findings

New formula for characteristic polynomials of deformed arrangements

Simplified proofs of existing results

Extension of results to type B arrangements

Abstract

We obtain a novel formula for characteristic polynomials of deformations of the Braid arrangement using the notion of levels of regions. As an application, we recover and strengthen results of Chen et al. on the characteristic polynomial of several specific types of hyperplane arrangements via much simpler arguments. Our theorem also generalizes to type B.