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.

The Allcock ball quotient

“…Indeed it is easy to check that the critical subdiagrams are the connected parabolic diagrams of type A 5 or D 4 . Since N (A 5 ) = 3A 5 and N (D 4 ) = 4D 4 are both parabolic and have both Gram matrices of rank 12 this follows from the theorem of Vinberg. The 26-cell P has two natural vertices w P perpendicular to all e p with p ∈ P and w L perpendicular to all e l with l ∈ L. The midpoint w 0 on the geodesic from w P to w L is called the Weyl point.…”

The complex Lorentzian Leech lattice and the bimonster

“…For example, by the above definition, the squared height is a positive element of Z[ √ 3], so it is not even a priori obvious that all the roots have height greater than or equal to one. What is more intriguing is that all the above remarks hold true for the example of the quaternionic lattice too (see [4]). …”

“…(First check this for i = 3, 4, 6; in these cases the algorithm leads to one of the simple roots showing these generators are in rad(D).) For some of the vectors ge[i] this algorithm gets stuck and then we perturbed it by reflecting in either ge [3],ge [4] or ge [6]. Which vector to perturb by is given in the list perturb.…”

Modular lattices from finite projective planes

“…The lattice obtained in this case is a direct sum of a quaternionic form of the Leech lattice and a hyperbolic cell. The reflection group of this lattice has properties analogous to the reflection group of the lattice L 3 mentioned in (2); see [6].…”

“…(Of course in this case one has to be careful to phrase everything in terms of right modules or left modules.) The reflection group of L H 2 was studied in [6] where we always considered right H-modules. One checks that z = w P + w Lp (−1 + i + j + k)/2 is a primitive null vector in L H 2 .…”