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Abstract

We study the Clebsch–Gordan problem for quiver representations, i.e. the problem of decomposing the point-wise tensor product of any two representations of a quiver into its indecomposable direct summands. For this purpose we develop results describing the behaviour of the point-wise tensor product under Galois coverings. These are applied to solve the Clebsch–Gordan problem for the double loop quivers with relations $\alpha\beta = \beta\alpha = \alpha^n = \beta^n=0$. These quivers were originally studied by I. M. Gelfand and V. A. Ponomarev in their investigation of representations of the Lorentz group. We also solve the Clebsch–Gordan problem for all quivers of type $\tilde{\mathbb{A}}_n$.