On the representation ring of a quiver.
The classical Clebsch-Gordan problem is to give a formula for decomposition of tensor product of two indecomposable representations of a group. In this paper a version of this problem is treated. Instead of representations of a group the representations of a quiver are considered and the tensor product is defined point-wise and arrow-wise. Fix a quiver \(Q\) and a field \(k\). There is a ring structure on the Abelian group freely generated by isomorphism classes of indecomposable (finite dimensional) \(k\)-representations of \(Q\). The addition (resp. multiplication) is induced by direct sum (resp. tensor product). The ring is called the representation ring of \(Q\) and denoted by \(R(Q)\). It is shown that if \(Q\) is a Dynkin quiver then \(R(Q)\) is a commutative \(\mathbb{Z}\)-order. The Clebsch-Gordan problem is solved for quivers of type \(\mathbb{A}_n\) and \(\mathbb{D}_n\) and an explicit description of the representation ring is given in this case. It is obtained by applying so called characteristic representations and Clebsch-Gordan formula for such representations valid for arbitrary quiver and any ground field. For other results on the Clebsch-Gordan problem for quiver representations see e.g. \textit{M. Herschend} [J. Algebra Appl. 4, No. 5, 481-488 (2005; Zbl 1082.16020)].
scientific article; zbMATH DE number 5659705
scientific article