Finite orbits of the braid group action on sets of reflections

TL;DR

This paper investigates the finite orbits of the braid group acting on sets of reflections, classifying them for finite Coxeter groups and analyzing invariants of reflection arrangements.

Contribution

It introduces a new method using universal generators to classify braid group orbits on reflections in finite groups, including Coxeter groups.

Findings

Classified orbits for non-degenerate arrangement matrices.

Determined all invariants of reflection arrangements.

Introduced universal generator sets for finite Coxeter groups.

Abstract

The finite orbits of the braid group action on Stokes matrices are studied and are shown to be the orbits on ordered sets of reflections, generating finite groups. All invariants of a reflection arrangement are determined. Determination of the orbits of the braid group on non-redundant generaing reflections in finite groups is done in a new way. The original idea is to introduce universal sets of generators in each group. These sets are found for all finite Coxeter groups. The orbits are classified for non-degenerate arrangement matrices.