Tensor products on quiver representations

The aim of this article is to translate the well-known tensor product of representations of a group given by diagonal action to the case of representations of a quiver. We provide three different approaches and exhibit their close relationship to the point-wise tensor product, which is considered in [M. Herschend, Solution to the Clebsch–Gordan problem for Kronecker representations. U.U.D.M Project Report 2003:P1, Uppsala University, 2003; M. Herschend, Solution to the Clebsch–Gordan problem for representations of quivers of type image, J. Algebra Appl. 4 (5) (2005) 481–488; M. Herschend, On the Clebsch–Gordan problem for quiver representations. U.U.D.M Report 2005:43, Uppsala University, 2005].