Leech Roots and Vinberg Groups

Abstract

This chapter investigates the remarkable properties of the Leech roots, which are the fundamental roots for the even unimodular lattice in Lorentzian space R ^25,1, and correspond one for one with the points of the Leech lattice. We give an extensive table of the Leech roots in both Euclidean and hyperbolic coordinates. We also provide the first of what promise to be many applications by showing that the Leech roots simplify and explain the remarkable results of Vinberg, Kaplinskaja and Meyer on the reflection groups of unimodular Lorentzian lattices in dimensions below 20. They also enable us to make some progress on the study of these groups in the next few dimensions.