Hyperbolic reflection groups associated to the quadratic forms $-3x_0^2 + x_1^2 + ... + x_n^2$

TL;DR

This paper classifies the maximal hyperbolic reflection groups linked to specific quadratic forms, detailing their structures up to dimension 13 and proving non-reflectivity in higher dimensions.

Contribution

It explicitly determines the Coxeter schemes of fundamental polyhedra for these groups and establishes the dimensional limit for their existence.

Findings

Hyperbolic reflection groups exist up to dimension 13.

Coxeter schemes of fundamental polyhedra are explicitly described.

Quadratic forms are non-reflective in dimensions higher than 13.

Abstract

We determine the maximal hyperbolic reflection groups associated to the quadratic forms $-3x_0^2 + x_1^2 + ... + x_n^2$, $n \ge 2$, and present the Coxeter schemes of their fundamental polyhedra. These groups exist in dimensions up to 13, and a proof is given that in higher dimensions these quadratic forms are not reflective.