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THE REFLECTIVE LORENTZIAN LATTICES OF DANIEL ALLCOCK
 

Summary: THE REFLECTIVE LORENTZIAN LATTICES OF
RANK 3
DANIEL ALLCOCK
Abstract. We classify all the symmetric integer bilinear forms of
signature (2, 1) whose isometry groups are generated up to finite
index by reflections. There are 8595 of them up to scale, whose 374
distinct Weyl groups fall into 39 commensurability classes. This ex-
tends Nikulin's enumeration of the strongly square-free cases. Our
technique is an analysis of the shape of the Weyl chamber, followed
by computer work using Vinberg's algorithm and our "method of
bijections". We also correct a minor error in Conway and Sloane's
definition of their canonical 2-adic symbol.
1. Introduction
Lorentzian lattices, that is, integral symmetric bilinear forms of sig-
nature (n, 1), play a major role in K3 surface theory and the structure
theory of hyperbolic Kac-Moody algebras. In both cases, the lattices
which are reflective, meaning that their isometry groups are generated
by reflections up to finite index, play a special role. In the KM case
they provide candidates for root lattices of KMA's with hyperbolic
Weyl groups that are large enough to be interesting. For K3's they

  

Source: Allcock, Daniel - Department of Mathematics, University of Texas at Austin

 

Collections: Mathematics