A note on Auslander-Reiten quivers for integral group rings
Let \(\mathcal O\) be a complete discrete valuation ring with residue field \(F\) of characteristic \(p > 0\), and let \(R \in \{{\mathcal O}, F\}\). Let \(G\) be a finite group, and let \(\Theta\) be a connected component of the stable Auslander-Reiten quiver \(\Gamma_s(RG)\) for the group algebra \(RG\). Let \(Q\) be a minimal element in the set of vertices of indecomposable \(RG\)-lattices in \(\Theta\), let \(M\) be an indecomposable \(RG\)-lattice in \(\Theta\) with vertex \(Q\), and let \(N\) be the Green correspondent of \(M\) in \(H := N_G(Q)\). The authors show that there is a graph monomorphism \(\gamma\) from \(\Theta\) into the connected component \(\Delta\) of \(\Gamma_s(RH)\) containing \(N\) which preserves edge-multiplicity and direction. This extends to \(\mathcal O\) a result of \textit{S. Kawata} [Osaka J. Math. 26, No. 3, 671-678 (1989; Zbl 0705.20009)] who worked over \(F\). The authors show also that \(\gamma\) is not necessarily an isomorphism.
scientific article; zbMATH DE number 880184