Reflective Integral Lattices

✍ Scribed by Rudolf Scharlau; Britta Blaschke

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 280 KB
• Volume: 181
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1996.0155

✦ Synopsis

A lattice L with a positive definite quadratic form is called reflective if the Ž . unique largest subgroup generated by reflections of the orthogonal group O L has Ž . no fixed vector. Equivalently, the ''root system'' R L has maximal rank. The root system of a lattice is defined in Section 1; the roots are not necessarily of length 1 or 2. In Section 2, the structure of reflective lattices is worked out. They are Ž . described and classified by pairs R, L L , where R is a ''scaled root system'' and the Ž . ''code'' L L is a subgroup of the ''reduced discriminant group'' T R . The crucial Ž . point is that T R only depends on the combinatorial equivalence class of the root system R. In Section 3, we give a precise description of the full root system of a reflective lattice if one starts with a sub-root-system of combinatorial type nA or 1 mA . In Section 4, our techniques are applied to a complete and explicit descrip-2 tion of all reflective lattices in dimensions F 6.