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 αβ = βα = αⁿ = βⁿ = 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 𝔸̃ₙ.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm109-2-3,
author = {Martin Herschend},
title = {Galois coverings and the Clebsch-Gordan problem for quiver representations},
journal = {Colloquium Mathematicae},
volume = {107},
year = {2007},
pages = {193-215},
zbl = {1167.16010},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm109-2-3}
}
Martin Herschend. Galois coverings and the Clebsch-Gordan problem for quiver representations. Colloquium Mathematicae, Tome 107 (2007) pp. 193-215. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm109-2-3/