Reflection group of the quaternionic Lorentzian Leech lattice

Abstract

We study a second example of the phenomenon studied in the article “The complex Lorentzian Leech lattice and the bimonster.” We find 14 roots in the automorphism group of the quaternionic Lorentzian Leech lattice L that form the Coxeter diagram given by the incidence graph of projective plane over F 2 . We prove that the reflections in these roots generate the automorphism group of L. The investigation is guided by an analogy with the theory of Weyl groups. There is a unique point in the quaternionic hyperbolic space fixed by the “diagram automorphisms” that we call the Weyl vector. The unit multiples of the 14 roots forming the diagram are the analogs of the simple roots. The 14 mirrors perpendicular to the simple roots are the mirrors that are closest to the Weyl vector.