# Inequalities for free multi-braid arrangements

## TL;DR

This paper characterizes free multiplicities in free multi-braid arrangements within a large cone, confirming previous classifications and proposing a new conjecture on their structure.

## Contribution

It proves that the only free multiplicities in a large cone are those previously identified and introduces a conjecture on the overall structure of free multiplicities.

## Key findings

- Confirmed classification of free multiplicities in a large cone
- Identified that only known free multiplicities occur in this region
- Proposed a conjecture on the structure of all free multiplicities

## Abstract

We prove that, on a large cone containing the constant multiplicities, the only free multiplicities on the braid arrangement are those identified in work of Abe, Nuida, and Numata (2009). We also give a conjecture on the structure of all free multiplicities on braid arrangements.