The Auslander-Reiten quiver of a modular group algebra revisited
Let G be a finite group, F a field of characteristic \(p>0\). In an important paper [Math. Z. 179, 97-121 (1982; Zbl 0479.20008)] \textit{P. Webb} proved that the tree class of a connected component of the stable Auslander-Reiten quiver of the group algebra FG is either a Dynkin diagram \(A_ n\), a Euclidean diagram or one of the infinite trees \(A_{\infty}\), \(B_{\infty}\), \(C_{\infty}\), \(D_{\infty}\), \(A^{\infty}_{\infty}\). For \(p\neq 2\), \textit{T. Okuyama} [J. Algebra 110, 425-430 (1987; Zbl 0624.20006)] proved that the tree class is never a Euclidean diagram. In this paper it is shown that for \(p=2\) only the Euclidean diagrams \(\tilde A_{12}\) or \(\tilde B_ 3\) are possible; it is known that these diagrams occur for \(G={\mathbb{Z}}_ 2\times {\mathbb{Z}}_ 2\) resp. \(G=A_ 4\) and \(F=GF(2)\). More precisely, it is proved that the block containing such a Euclidean component has a Klein four group as a defect group. For the reduced graph of a component also the Euclidean type \(\tilde A_ n\) is considered.
scientific article; zbMATH DE number 4134292