“…By a similar technique, for a quiver Q with relations in R and a finite abelian group G ⊆ Aut(Q) preserving the relations in R, we gave in [8] the condition for a (Q, R)-representation to be a Λ * G-module and determined the number of non-isomorphic indecomposable Λ * G-modules which are induced from the same (Q, R)-representation, where Λ = kQ/ R . In the paper [9], we discussed the duality of (Q, G) in the case that G ⊆ Aut(kQ) is finite abelian, and by the duality, gave the correspondence between the indecomposable Q-representations and the positive roots of the valued graph of (Q, G). In this paper, we consider the duality of the Auslander-Reiten quiver of kQ.…”