Reflective integral lattices
The root system \(R(L)\) of a positive definite integral lattice \(L\) consists of the primitive vectors in \(L\) whose associated hyperplane reflection maps \(L\) into itself. Following \textit{E. B. Vinberg} [Math. USSR, Sb. 16, 17-35 (1972); translation from Mat. Sb., Nov. Ser. 87(129), 18-36 (1972; Zbl 0244.20058)], the authors call \(L\) reflective if the sublattice generated by \(R(L)\) has the same rank as \(L\). Reflective lattices are constructed and classified by pairs \((R, C)\) where \(R\) is a ``scaled root system'' and \(C\) a ``glue code'', i.e. some subgroup of \(\langle R\rangle^*/ \langle R\rangle\). The first step in the general classification process is to determine an appropriate set of root systems \(R_0\) of rank \(n\) such that for each indecomposable \(n\)-dimensional reflective lattice \(L\) the root system \(R(L)\) contains \(R_0\). Main results of the paper are descriptions of \(R (L)\) when \(R_0\) is of type \(nA_1\) or \(mA_2\), and the classification of all reflective lattices in dimensions \(n\leq 6\).
scientific article; zbMATH DE number 883795