Reflection group of the quaternionic Lorentzian Leech lattice

Abstract

In this article we study a second example of the phenomenon studied in "Complex Lorentzian Leech lattice and bimonster".(Arxiv. math.GR/0508228). The results and methods of proof are similar. We find 14 roots in the automorphism group of the the quaternionic Lorentzian Leech lattice L that form the Coxeter diagram D given by the incidence graph of projective plane over finite field of order 2. We prove that the reflections in these 14 roots generate the automorphism group of L. We find evidence that these reflections behave like the simple roots and the vector fixed by the diagram automorphisms behaves like the Weyl vector for the reflection group. Much of the work follows the analogy that D is like the Coxeter-Dynkin diagram for this hyperbolic reflection group.