Galois coverings and the Clebsch–Gordan problem for quiver representations

Martin Herschend¹

¹ Department of Mathematics Uppsala University, Box 480 SE-751 06 Uppsala, Sweden

Colloquium Mathematicum, Tome 109 (2007) no. 2, pp. 193-215

Résumé

We study the Clebsch–Gordan problem for quiver representations, i.e. the problem of decomposing the point-wise tensor product of any two representations of a quiver into its indecomposable direct summands. For this purpose we develop results describing the behaviour of the point-wise tensor product under Galois coverings. These are applied to solve the Clebsch–Gordan problem for the double loop quivers with relations $\alpha\beta = \beta\alpha = \alpha^n = \beta^n=0$. These quivers were originally studied by I. M. Gelfand and V. A. Ponomarev in their investigation of representations of the Lorentz group. We also solve the Clebsch–Gordan problem for all quivers of type $\tilde{\mathbb{A}}_n$.

Détail

DOI : 10.4064/cm109-2-3

Keywords: study clebsch gordan problem quiver representations problem decomposing point wise tensor product representations quiver its indecomposable direct summands purpose develop results describing behaviour point wise tensor product under galois coverings these applied solve clebsch gordan problem double loop quivers relations alpha beta beta alpha alpha beta these quivers originally studied nbsp gelfand nbsp nbsp ponomarev their investigation representations lorentz group solve clebsch gordan problem quivers type tilde mathbb