On the representation ring of a quiver.
type: Item
instance of: scholarly article
title: On the representation ring of a quiver.
zbMATH Open document ID: 1197.16018
author: Martin Herschend
published in: Algebras and Representation Theory
publication date: 2010-01-18T00:00:00Z
review text: The classical Clebsch-Gordan problem is to give a formula for decomposition of tensor product of two indecomposable representations of a group. In this paper a version of this problem is treated. Instead of representations of a group the representations of a quiver are considered and the tensor product is defined point-wise and arrow-wise. Fix a quiver \(Q\) and a field \(k\). There is a ring structure on the Abelian group freely generated by isomorphism classes of indecomposable (finite dimensional) \(k\)-representations of \(Q\). The addition (resp. multiplication) is induced by direct sum (resp. tensor product). The ring is called the representation ring of \(Q\) and denoted by \(R(Q)\). It is shown that if \(Q\) is a Dynkin quiver then \(R(Q)\) is a commutative \(\mathbb{Z}\)-order. The Clebsch-Gordan problem is solved for quivers of type \(\mathbb{A}_n\) and \(\mathbb{D}_n\) and an explicit description of the representation ring is given in this case. It is obtained by applying so called characteristic representations and Clebsch-Gordan formula for such representations valid for arbitrary quiver and any ground field. For other results on the Clebsch-Gordan problem for quiver representations see e.g. \textit{M. Herschend} [J. Algebra Appl. 4, No. 5, 481-488 (2005; Zbl 1082.16020)].
reviewed by: Stanisław Kasjan
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 19A22
Mathematics Subject Classification ID: 15A69
zbMATH DE Number: 5659705
zbMATH Keywords: representation rings
zbMATH Keywords: quiver representations
zbMATH Keywords: Dynkin quivers
zbMATH Keywords: Clebsch-Gordan problem
zbMATH Keywords: tensor products
MaRDI profile type: Publication
full work available at URL: s10468-008-9118-1
OpenAlex ID: W2057145869
cites work: Lattices over curve singularities with large conductor
cites work: Algebra VIII: Representations of finite-dimensional algebras. Transl. from the Russian
cites work: SOLUTION TO THE CLEBSCH–GORDAN PROBLEM FOR REPRESENTATIONS OF QUIVERS OF TYPE $\tilde{\mathbb{A}}_n$
cites work: Q3997012
cites work: Algebra I. Basic notions of algebra. Transl. from the Russian by M. Reid
DOI: 10.1007/S10468-008-9118-1
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description: scientific article; zbMATH DE number 5659705