#### Vol. 6, No. 5, 2012

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Idempotents in representation rings of quivers

### Ryan Kinser and Ralf Schiffler

Vol. 6 (2012), No. 5, 967–994
##### Abstract

For an acyclic quiver $Q$, we solve the Clebsch–Gordan problem for the projective representations by computing the multiplicity of a given indecomposable projective in the tensor product of two indecomposable projectives. Motivated by this problem for arbitrary representations, we study idempotents in the representation ring of $Q$ (the free abelian group on the indecomposable representations, with multiplication given by tensor product). We give a general technique for constructing such idempotents and for decomposing the representation ring into a direct product of ideals, utilizing morphisms between quivers and categorical Möbius inversion.

##### Keywords
quiver, representation ring, tensor product, idempotents
##### Mathematical Subject Classification 2010
Primary: 16G20
Secondary: 19A22, 06A99