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On the representation ring of a quiver
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
2009 (English)In: Algebras and Representation Theory, ISSN 1386-923X, E-ISSN 1572-9079, Vol. 12, no 6, 513-541 p.Article in journal (Refereed) Published
Abstract [en]

The Clebsch-Gordan problem for quiver representations is the problem of   decomposing the tensor product of any two representations of a quiver,   where the tensor product is defined point-wise and arrow-wise. We   introduce the so-called characteristic representations and decompose   their tensor product. This is then applied to solve the Clebsch-Gordan   problem for quivers of type A(n) and D-n. In both cases we also provide   an explicit description of the representation ring.

Place, publisher, year, edition, pages
2009. Vol. 12, no 6, 513-541 p.
Keyword [en]
Dynkin quivers, Quiver representations, Tensor product, Clebsch-Gordan problem, Representation ring
National Category
Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-97101DOI: 10.1007/s10468-008-9118-1ISI: 000271527000002OAI: oai:DiVA.org:uu-97101DiVA: diva2:171895
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
Opponent
Supervisors
Available from: 2008-04-29 Created: 2008-04-29 Last updated: 2016-04-29Bibliographically approved

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Herschend, Martin

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