Hyperbolic reflection groups associated to the quadratic forms $${-3x_0^2 + x_1^2 + \cdots + x_n^2}$$

Abstract

We determine the maximal hyperbolic reflection groups associated to the quadratic forms \({-3x_0^2 + x_1^2 + \cdots + x_n^2, n \ge 2}\), 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.