On Auslander-Reiten components for certain group modules (Q1324959)

Let \(F\) be an algebraically closed field of prime characteristic \(p\), let \(G\) be a finite group, and let \(P\) be a Sylow \(p\)-subgroup of \(G\). The author investigates the structure of a component \(\Theta\) of the stable Auslander-Reiten quiver of the group algebra \(FG\) containing a module \(M\) of dimension prime to \(p\). Suppose that \(P\) is neither cyclic nor of order 4 nor (generalized) quaternion. Then \(\Theta\) is isomorphic to the connected component \(\Theta_ 0\) containing the trivial module, and \(P\) is a vertex of every module in \(\Theta_ 0\). The author presents examples where the isomorphism \(\Theta \simeq \Theta_ 0\) is given by tensoring with \(M\).