Reflective Integral Lattices Original Research Article

A latticeLwith a positive definite quadratic form is called reflective if the unique largest subgroup generated by reflections of the orthogonal group O(L) has 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 described and classified by pairs (R, image), where R is a “scaled root system” and the “code” image is a subgroup of the “reduced discriminant group” T(R). The crucial 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 typenA1ormA2. In Section 4, our techniques are applied to a complete and explicit description of all reflective lattices in dimensions ≤6.