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
Milestones
Received: 9 September 2010
Revised: 3 September 2011
Accepted: 3 October 2011
Published: 31 July 2012
Authors
Ryan Kinser
Department of Mathematics
Northeastern University
360 Huntington Avenue
Boston 02213
United States
http://www.math.neu.edu/~rkinser/index.html
Ralf Schiffler
Department of Mathematics
University of Connecticut
196 Auditorium Rd
Unit 3009
Storrs, CT 06269
United States
http://www.math.uconn.edu/~schiffler/