Abstract
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 \(\mathbb{A}_n\) and \(\mathbb{D}_n\). In both cases we also provide an explicit description of the representation ring.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Dieterich, E.: Lattices over curve singularities with large conductor. Invent. Math. 114(2), 399–433 (1993), ISSN 0020-9910
Gabriel, P., Roiter, A.V.: Representations of finite-dimensional algebras. In: Algebra, VIII, Encyclopaedia of Mathematical Sciences, vol. 73, pp. 1–177. Springer, Berlin (1992), with a chapter by B. Keller
Herschend, M.: Solution to the Clebsch-Gordan problem for Kronecker representations. U.U.D.M Project report 2003:P1, Uppsala University (2003)
Herschend, M.: Solution to the Clebsch-Gordan problem for representations of quivers of type \(\tilde{A}_n\). J. Algebra Appl. 4(5), 481–488 (2005)
Huppert, B.: Angewandte lineare Algebra. viii+646 pp., Walter de Gruyter, Berlin (1990), ISBN 3-11-012107-7
Shafarevich, I.R.: Basic notions of algebra. In: Algebra. I, Encyclopaedia of Mathematical Sciences (translated from the Russian by M. Reid), vol. 11, pp. iv–258. Springer, New York (1990), ISBN 3-540-17006-5
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Herschend, M. On the Representation Ring of a Quiver. Algebr Represent Theor 12, 513–541 (2009). https://doi.org/10.1007/s10468-008-9118-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10468-008-9118-1