Idempotents in representation rings of quivers

TL;DR

This paper investigates the structure of the representation ring of acyclic quivers, providing methods to construct idempotents and decompose the ring, thereby advancing understanding of tensor products of indecomposable representations.

Contribution

It introduces a general technique for constructing idempotents and decomposing the representation ring of quivers using morphisms and Moebius inversion, addressing the Clebsch-Gordan problem.

Findings

Computed multiplicities of indecomposable projectives in tensor products.

Developed a method to decompose the representation ring into a direct product of ideals.

Provided a framework for understanding tensor products in representation theory of quivers.

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.