Reflective Lorentzian lattices

The archive Lorentz.zip contains a list of all genera of strongly squarefree reflective Lorentzian lattices of signature (n,1) for n>=3.

The case n=4 is due to C. Walhorn and the case n=5 is due to I. Turkalj. These lists are complete.

For n>5, the lists have been compiled by R. Scharlau & M.Kirschmer and they are complete under the assumption that the determinants of such lattices have no prime divisors >19. We will prove this assumption in the upcoming paper "Classification of isotropic reflective Lorentzian lattices in higher dimensions".

For n=3, the list contains the strongly squarefree isotropic and anisotropic lattices. We will publish a proof of its completeness in an upcoming paper.

How to read these files

For each genus G of squarefree reflective lattices L of signature (n,1), the file "Listn" contains a single entry of the form < -det, symbol, #faces, #cusps >
where

det = determinant of G
symbol = Conway & Sloane genus symbol of G
faces[k] = number of n-k-dimensional faces of the fundamental domain of the reflection group W(L) of L. In particular, faces[1] is the number of vectors found by Vinberg's algorithm.
cusps = number of cusps of the fundamental domain

For each such genus G, the class number is one and the file "Gramn" contains a line with the (n+1)x(n+1) entries of a Gram matrix of L. The ordering of the genera in the files "Listn" and "Gramn" is the same.