The counting formula for indecomposable modules over string algebra

Abstract

Let $ A = kQ/I $ be a string algebra. We show that, if for any vertex $ v $ of its bound quiver $ (Q, I) $, there exists at most one arrow (resp. at most two arrows) ending with $ v $ and there exist at most two arrows (resp. at most one arrow) starting with $ v $, then the number of indecomposable modules over $ A $ is $ \dim_{k}A+\Sigma $, where $ \Sigma $ is induced by $ rad P(v) $ (resp. $ E(v)/\mathrm{soc} E(v) $) with decomposable socle (resp. top), where $ P(v) $ (resp. $ E(v) $) is the indecomposable projective (resp. injective) module corresponded by the vertex $ v $.

Keywords

• representation-finite
• representations of quivers
• indecomposable modules