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On the representation rings of quivers of exceptional Dynkin type
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
2008 (English)In: Bulletin des Sciences Mathématiques, ISSN 0007-4497, Vol. 132, no 5, 395-418 p.Article in journal (Refereed) Published
Abstract [en]

The direct sum and tensor product (defined point-wise and arrow-wise) of representations of a given quiver is encoded in the representation ring of that quiver. We provide methods which reduce the computation of certain parts of the representation ring of a quiver to that of connected subquivers. These methods, combined with known results on the representation rings of quivers of type A and D, and supplemented by certain matrix calculations, yield an explicit description of the representation rings of all quivers of type E-6.

Place, publisher, year, edition, pages
2008. Vol. 132, no 5, 395-418 p.
Keyword [en]
exceptional Dynkin quivers, quiver representations, tensor product; Clebsch-Gordan problem, representation ring
National Category
Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-97102DOI: 10.1016/j.bulsci.2007.11.005ISI: 000257536700003OAI: oai:DiVA.org:uu-97102DiVA: diva2:171896
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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