Shapes of Connected Components of the Auslansder-Reiten Quivers of Artin Algebras

The aim of these notes is to report some new developments on the problem of describing all possible shapes of the connected components of the Auslander-Reiten quiver ΓA of an artin algebra A. The problem is interesting since the shapes of these components carry some important information of the module category of A. For instance the algebra A is hereditary if and only if ΓA has a connected component of shape N∆ where ∆ is a quiver without oriented cycles such that the number of its vertices is the same as that of simple A-modules. More importantly, by analyzing the structure of Auslander-Reiten components, Riedtmann classified the self-injective algebras of finite representation type [44, 45, 46], and Erdmann did the same for the blocks of finite groups with a dihedral or semidihedral defect group. And remarkably Erdmann has recently showed that the representation type of a block of a finite group is determined by the shapes of the connected components of its Auslander-Reiten quiver [26]. More generally in any preprojective or preinjective Auslander-Reiten components, modules are determined by their composition factors and the maps are sums of composites of irreducible maps [29]. Furthermore modules in a quasi-serial Auslander-Reiten component behave like serial modules [47].