Additively indecomposable positive integral quadratic forms
Let \(L \cong \mathbb{Z}^ n\) be a lattice. The set \(\text{Bil}_{\geq 0} (L)\) of positive semidefinite integral bilinear forms on \(L\) is a semigroup under addition on which \(\Aut (L) \cong \text{GL}_ n(\mathbb{Z})\) acts. A form \(\varphi\) is called additively decomposable if \(\varphi\) can be written as a sum of two nonzero elements of \(\text{Bil}_{\geq 0} (L)\). By a classical result of Mordell there are only finitely many isometry classes of positive semidefinite additively indecomposable \(n\)- dimensional lattices. The author gives a new proof of this result. He further classifies all such forms up to dimension \(n=8\). Now let \(G\) be a finite group and let \(L\) be a \(\mathbb{Z} G\)-lattice. \(\text{Bil}^ G_{\geq 0} (L)\) consists of all \(G\)-invariant elements of \(\text{Bil}_{\geq 0} (L)\). Then up to \(\Aut_{\mathbb{Z} G} (L)\)-action there are only finitely many additively \((G)\)-indecomposable forms in \(\text{Bil}^ G_{\geq 0} (L)\).
additively indecomposable \(n\)-dimensional lattices
scientific article; zbMATH DE number 599845