Solution to the Clebsch–Gordan problem for string algebras

The category of modules over a string algebra is equipped with a tensor product defined point-wise and arrow-wise in terms of the underlying quiver. In the present article we investigate how this tensor product interacts with the classification of indecomposables. We apply the results obtained to solve the Clebsch–Gordan problem for string algebras. Moreover, we describe the corresponding representation ring and tensor ideals in the module category.