# Freeness of multi-reflection arrangements via primitive vector fields

## TL;DR

This paper extends Terao's freeness results for reflection arrangements to more general unitary reflection groups, linking exponents to dual and Galois-twisted representations, and broadening understanding of hyperplane arrangement freeness.

## Contribution

It generalizes Terao's freeness theorem to well-generated and imprimitive unitary reflection groups, introducing new connections with dual and Galois-twisted representations.

## Key findings

- Freeness of arrangements for well-generated unitary reflection groups.
- Explicit formulas for exponents involving dual and Galois-twisted representations.
- Extension of results to all imprimitive irreducible unitary reflection groups.

## Abstract

In 2002, Terao showed that every reflection multi-arrangement of a real reflection group with constant multiplicity is free by providing a basis of the module of derivations. We first generalize Terao's result to multi-arrangements stemming from well-generated unitary reflection groups, where the multiplicity of a hyperplane depends on the order of its stabilizer. Here the exponents depend on the exponents of the dual reflection representation. We then extend our results further to all imprimitive irreducible unitary reflection groups. In this case the exponents turn out to depend on the exponents of a certain Galois twist of the dual reflection representation that comes from a Beynon-Lusztig type semi-palindromicity of the fake degrees.