A colourful classification of (quasi) root systems and hyperplane arrangements

TL;DR

This paper introduces a graph-based classification scheme for root systems and hyperplane arrangements, providing a unified framework that generalizes existing structures and identifies unique noncrystallographic arrangements.

Contribution

It presents a novel graph-based approach to classify root systems and hyperplane arrangements, including noncrystallographic cases and subsystems, extending previous classification methods.

Findings

Classified root subsystems using coloured graphs.

Identified a unique noncrystallographic hyperplane arrangement.

Provided elementary classifications of closed and Levi root subsystems.

Abstract

We introduce a class of graphs with coloured edges to encode subsystems of the classical root systems, which in particular classify them up to equivalence. We further use the graphs to describe root-kernel intersections, as well as restrictions of root (sub)systems on such intersections, generalising the regular part of a Cartan subalgebra. We also consider a slight variation to encode the hyperplane arrangements only, showing there is a unique noncrystallographic arrangement that arises. Finally, a variation of the main definition leads to elementary classifications of closed and Levi root subsystems.