On the Representation Ring of a Quiver

Abstract

The Clebsch–Gordan problem for quiver representations is the problem of decomposing the tensor product of any two representations of a quiver, where the tensor product is defined point-wise and arrow-wise. We introduce the so-called characteristic representations and decompose their tensor product. This is then applied to solve the Clebsch–Gordan problem for quivers of type $\mathbb{A}_n$ and $\mathbb{D}_n$ . In both cases we also provide an explicit description of the representation ring.