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Galois coverings and the Clebsch-Gordan problem for quiver representations
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
2007 (English)In: Colloquium Mathematicum, ISSN 0010-1354, E-ISSN 1730-6302, Vol. 109, no 2, 193-215 p.Article in journal (Refereed) Published
##### Abstract [en]

We study the Clebsch--Gordan problem for quiverrepresentations, i.e. the problem of decomposing the point-wisetensor product of any two representations of a quiver into itsindecomposable direct summands. For this purpose we developresults describing the behaviour of the point-wise tensorproduct under Galois coverings. These are applied to solve theClebsch--Gordan problem for the double loop quivers withrelations $\alpha\beta = \beta\alpha = \alpha^n = \beta^n=0$.These quivers were originally studied by I. M. Gelfand and V. A.Ponomarev in their investigation of representations of theLorentz group. We also solve the Clebsch--Gordan problem for allquivers of type $\tilde{\mathbb{A}}_n$.

##### Place, publisher, year, edition, pages
2007. Vol. 109, no 2, 193-215 p.
Mathematics
##### Identifiers
OAI: oai:DiVA.org:uu-97099DiVA: diva2:171893
Available from: 2008-04-29 Created: 2008-04-29 Last updated: 2016-04-29Bibliographically approved
##### In thesis
1. On the Clebsch-Gordan problem for quiver representations
Open this publication in new window or tab >>On the Clebsch-Gordan problem for quiver representations
2008 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

On the category of representations of a given quiver we define a tensor product point-wise and arrow-wise. The corresponding Clebsch-Gordan problem of how the tensor product of indecomposable representations decomposes into a direct sum of indecomposable representations is the topic of this thesis.

The choice of tensor product is motivated by an investigation of possible ways to modify the classical tensor product from group representation theory to the case of quiver representations. It turns out that all of them yield tensor products which essentially are the same as the point-wise tensor product.

We solve the Clebsch-Gordan problem for all Dynkin quivers of type A, D and E6, and provide explicit descriptions of their respective representation rings. Furthermore, we investigate how the tensor product interacts with Galois coverings. The results obtained are used to solve the Clebsch-Gordan problem for all extended Dynkin quivers of type Ãn and the double loop quiver with relations βα=αβ=αnn=0.

##### Place, publisher, year, edition, pages
Uppsala: Matematiska institutionen, 2008. v, 34 p.
##### Series
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 56
##### Keyword
Algebra and geometry, quiver, quiver representation, tensor product, Clebsch-Gordan problem, representation ring, bialgebra, Galois covering, Algebra och geometri
##### National Category
Algebra and Logic Geometry
##### Identifiers
urn:nbn:se:uu:diva-8663 (URN)978-91-506-2002-3 (ISBN)
##### Public defence
2008-05-22, Häggsalen, Ångström Laboratory, Lägerhyddsvägen 1, Uppsala, 13:15
##### Supervisors
Available from: 2008-04-29 Created: 2008-04-29 Last updated: 2016-04-29Bibliographically approved

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