The modules induced from a normal subgroup and the Auslander-Reiten quiver
Let G be a finite group, let N be a normal subgroup of G and let F be a field of prime characteristic p. Denote by \(\theta\) a connected component of the stable Auslander-Reiten quiver \(\Gamma_ s(FG)\) of the group algebra FG. Assume also that there exist indecomposable N-projective FG- modules in \(\theta\). Choose an indecomposable N-projective FG-module \(L_ 0\) in \(\theta\), let \(S_ 0\) be an indecomposable FN-module with \(L_ 0\) a direct summand of \(S^ G_ 0\) and let T be the inertia group of a connected component of \(\Gamma_ s(FN)\) containing \(S_ 0\). Suppose that \(S^ T_ 0=U_ 0\oplus U_ 1\oplus...\oplus U_ n\) where \(U_ i\) is indecomposable and \(L_ 0\cong U^ G_ 0\). Let \(\Lambda\) be the connected component of \(\Gamma_ s(FT)\) containing \(U_ 0\). The main result of this paper asserts that if U is an indecomposable FT-module in \(\Lambda\), then the induced module \(U^ G\) is indecomposable and induction from T to G gives a graph isomorphism from \(\Lambda\) onto \(\theta\) which preserves edge-multiplicity and direction.
scientific article; zbMATH DE number 4150436