The normaliser action and strongly modular lattices
Let \(L\subseteq \mathbb{R}^d\) be an even integral lattice in the Euclidean space of dimension \(d\) and let \(L^{\#}\subseteq \mathbb{R}^d\) be its dual lattice. Let \(\pi(L)\) be the set of all intermediate lattices \(L\leq L'\leq L^{\#}\) that are inverse images of sums of Sylow subgroups of the finite abelian group \(L^{\#}/L\). Then, \(L\) is said to be strongly modular if \(L\) is similar to \(L'\) for all \(L'\in \pi(L)\) [cf. \textit{H. G. Quebbemann}, Enseign. Math., II. Sér. 43, 55-65 (1997; Zbl 0898.11014)]. Recall that \(L\) and \(L'\) are called similar if there exists \(s\in \text{GL}(\mathbb{R} L)\) and \(a\in\mathbb{R}_{>0}\) such that \(Ls= L'\) and \((vs,ws)= a(v, w)\) for all \(v,w\in \mathbb{R} L\), where \(( ,)\) denotes the Euclidean scalar product. The author derives group theoretical methods to test if a lattice is strongly modular and applies these methods to the lattices of rational irreducible maximal finite groups.
lattices of rational irreducible maximal finite groups
scientific article; zbMATH DE number 1089509