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arXiv · 1009.0029

Idempotents in representation rings of quivers

Ryan Kinser·Ralf Schiffler
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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 Moebius inversion.

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Ryan Kinser, Ralf Schiffler. 2012-08-25. Idempotents in representation rings of quivers. https://doi.org/10.2140/ant.2012.6.967

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