Module correspondence in Auslander-Reiten quivers for finite groups
The theory of almost split sequences due to M. Auslander and I. Reiten has also become a fundamental tool in the study of modular representations of finite groups G over fields K with characteristic \(p>0\). In this important paper the author presents several deep and useful results describing the relation between the Green correspondence and the restrictions of almost split sequences. Let \(\theta\) be a connected component of the stable Auslander-Reiten quiver \(\Gamma_ s(KG)\) of the group algebra KG. Let \(V(\theta)=\{vx(M)|\) M indecomposable KG-module in \(\theta\}\), where vx(M) denotes the vertex of M. In Proposition 3.2 the author shows that V(\(\theta\)) contains a minimal element Q with respect to the partial order \(\leq_ G\). Furthermore, Q is uniquely determined by \(\theta\) up to conjugation with elements of G. Fix such a p-subgroup Q, and let \(N=N_ G(Q)\). Choose an indecomposable KG-module \(M_ 0\) in \(\theta\), and let the KN-module \(f(M_ 0)\) be its Green correspondent. Let \(\Delta\) be the connected component of \(\Gamma_ s(KN)\) containing \(f(M_ 0)\). Then Theorem 4.6 asserts that there is a subquiver \(\Lambda\) of \(\Delta\) and a graph isomorphism \(\psi\) from \(\Lambda\) onto \(\theta\) which preserves edge-multiplicity and direction. Furthermore, \(\psi\) induces a one-to-one correspondence between the indecomposable KG-modules M in \(\theta\) and the indecomposable KN-modules in \(\Lambda\). As an immediate consequence the author shows in Corollary 4.7 that \(vx(M)\leq_ GN_ G(Q)\) for all indecomposable KG-modules M in \(\theta\).
scientific article; zbMATH DE number 4156747