Even More Infinite Ball Packings from Lorentzian Root Systems

TL;DR

This paper extends Maxwell's interpretation of Boyd's infinite ball packings using Lorentzian root systems, introducing a geometric 'level' concept that generalizes previous graph-based definitions and classifies certain level-2 root systems.

Contribution

It proposes a geometric 'level' for root systems, extending Maxwell's results to more general positively independent roots and classifies specific level-2 Coxeter polytopes.

Findings

Maxwell's results extend to general root systems with positive independence.

A geometric 'level' concept replaces the graph-theoretic 'level'.

Partial classification of level-2 Coxeter d-polytopes with d+2 facets.

Abstract

Boyd (1974) proposed a class of infinite ball packings that are generated by inversions. Later, Maxwell (1983) interpreted Boyd's construction in terms of root systems in Lorentz space. In particular, he showed that the space-like weight vectors correspond to a ball packing if and only if the associated Coxeter graph is of "level $2$." In Maxwell's work, the simple roots form a basis of the representations space of the Coxeter group. In several recent studies, the more general based root system is considered, where the simple roots are only required to be positively independent. In this paper, we propose a geometric version of "level" for the root system to replace Maxwell's graph theoretical "level." Then we show that Maxwell's results naturally extend to the more general root systems with positively independent simple roots. In particular, the space-like extreme rays of the Tits cone correspond to a ball packing if and only if the root system is of level $2$. We also present a partial classification of level-$2$ root systems, namely the Coxeter $d$-polytopes of level-$2$ with $d+2$ facets.