REFLECTIVE INTEGRAL LATTICES

A lattice L with a positive definite quadratic form is called reflecti ve if the unique largest subgroup generated by reflections of the orth ogonal 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 S ection 1; the roots are not necessarily of length 1 or 2. In Section 2 , the structure of reflective lattices is worked out. They are describ ed and classified by pairs (R,L), where R is a ''scaled root system'' and the ''code'' L is a subgroup of the ''reduced discriminant group'' (T) over bar(R). The crucial point is that (T) over bar(R) only depen ds on the combinatorial equivalence class of the root system R. In Sec tion 3, we give a precise description of the full root system of a ref lective lattice if one starts with a sub-root-system of combinatorial type nA(1) or mA(2). In Section 4, our techniques are applied to a com plete and explicit description of all reflective lattices in dimension s less than or equal to 6. (C) 1996 Academic Press, Inc.