Reflective Lorentzian lattices of signature \((5,1)\)
A \(\mathbb{Z}\)-lattice of signature \((n,m)\) is a pair \((L,b)\), where \(L\) is a free \(\mathbb{Z}\)-module of rank \(n+m\) equipped with a symmetric bilinear form \(b : L \times L \to \mathbb{Q}\) of signature \((n,m)\). By a Lorentzian lattice people usually mean \(\mathbb{Z}\)-lattices of signature \((n,1)\). These notions can also be extended to lattices over rings of integers of totally real number fields. A lattice \(E\) like this is called reflective if its isometry group \(O^+(E)\) is generated up to finite index by reflections. Further, a positive definite lattice is called totally-reflective if every lattice in its genus has a root system of full rank. The paper under review is concerned with classification of reflective lattices of signature \((5,1)\). The author shows that every such lattice can be written as orthogonal direct sum \(^{\alpha} \mathbb{H} \perp L\), where \(^{\alpha} \mathbb{H} \cong \begin{pmatrix} 0 & \alpha \\ \alpha & 0 \end{pmatrix}\) and \(L\) is totally-reflective. The author then uses this result to prove that up to isometry there are \(80\) strongly square-free (a certain technical local condition on Jordan decomposition blocks) reflective \(\mathbb{Z}\)-lattices of signature \((5, 1)\). The author is also interested in totally-reflective lattices as objects in their own right, and obtains some classification results for these positive definite lattices as well.
scientific article; zbMATH DE number 6942907
scientific article