On Auslander-Reiten components and height zero lattices for integral group rings.
Let \(G\) be a finite group, let \(k=k'\) be an algebraically closed field of characteristic \(p>0\), and let \((K,R,k)\) and \((K',R',k')\) be \(p\)-local systems such that \(R\) is a ramified extension of \(R'\) of ramification index at least \(3\). Then the author studies the possible structures of Auslander-Reiten components of \(RG\)-lattices \(L\) of height \(0\). Webb showed that only components of tree class \(A_\infty\), \(B_\infty\), \(C_\infty\), \(D_\infty\) and \(A_\infty^\infty\) or a Euclidean diagram can occur. The paper under review shows that in case of tree class \(A_\infty\) an indecomposable \(RG\)-lattice of height \(0\) lies at basically the same position as its source. Moreover, the author gives various criteria when the tree class of a component is \(A_\infty\). One of the main tools for the proof is Knörr's concept of virtually irreducible lattices.
integral representations of finite groups
scientific article; zbMATH DE number 6370054
scientific article