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The Reflective Lorentzian Lattices of Rank 3
Daniel Allcock, University of Texas at Austin, TX

Memoirs of the American Mathematical Society
2012; 108 pp; softcover
Volume: 220
ISBN-10: 0-8218-6911-6
ISBN-13: 978-0-8218-6911-6
List Price: US$70
Individual Members: US$42
Institutional Members: US$56
Order Code: MEMO/220/1033
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The author classifies all the symmetric integer bilinear forms of signature \((2,1)\) whose isometry groups are generated up to finite index by reflections. There are 8,595 of them up to scale, whose 374 distinct Weyl groups fall into 39 commensurability classes. This extends Nikulin's enumeration of the strongly square-free cases. The author's technique is an analysis of the shape of the Weyl chamber, followed by computer work using Vinberg's algorithm and a "method of bijections". He also corrects a minor error in Conway and Sloane's definition of their canonical \(2\)-adic symbol.

Table of Contents

  • Background
  • The classification theorem
  • The reflective lattices
  • Bibliography
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