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

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.

quiver, representation ring, tensor product, idempotents
Mathematical Subject Classification 2010
Primary: 16G20
Secondary: 19A22, 06A99
Received: 9 September 2010
Revised: 3 September 2011
Accepted: 3 October 2011
Published: 31 July 2012
Ryan Kinser
Department of Mathematics
Northeastern University
360 Huntington Avenue
Boston 02213
United States
Ralf Schiffler
Department of Mathematics
University of Connecticut
196 Auditorium Rd
Unit 3009
Storrs, CT 06269
United States