Representations of quivers with free module of covariants.
type: Item
instance of: scholarly article
title: Representations of quivers with free module of covariants.
zbMATH Open document ID: 1064.16014
author: Carol Chang
author: Jerzy Weyman
published in: Journal of Pure and Applied Algebra
publication date: 2004-08-19T00:00:00Z
review text: The isomorphism classes of representations with a fixed dimension vector \(\mathbf d\) of a quiver \(Q\) are in a bijective correspondence with the orbits of \(\text{GL}({\mathbf d})\), a product of general linear groups, acting linearly on the representation space \(R(Q,{\mathbf d})\). It is well known that for a Dynkin type quiver \(Q\) the ring \(\text{SI}(Q,{\mathbf d})\) of semi-invariant polynomial functions on \(R(Q,{\mathbf d})\) is a polynomial ring. In the present paper it is proved that if \(Q\) is of Dynkin type \(A_n\), then the ideal in the coordinate ring \(k[R(Q,{\mathbf d})]\) generated by the semi-invariants of positive degree is a complete intersection. As a corollary, when the base field is of characteristic zero, all modules of \(\text{SL}({\mathbf d})\)-covariants in \(k[R(Q,{\mathbf d})]\) are free modules over \(\text{SI}(Q,{\mathbf d})\). An example is presented showing that the corresponding statement does not hold for all Dynkin quivers. A modified statement valid for arbitrary tame quivers was proved by \textit{Ch. Riedtmann} and \textit{G. Zwara} [Comment. Math. Helv. 79, No. 2, 350-361 (2004; Zbl 1063.14052)]. (Also submitted to MR.)
reviewed by: Matyas Domokos
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 14L24
Mathematics Subject Classification ID: 14M10
Mathematics Subject Classification ID: 15A72
Mathematics Subject Classification ID: 16R30
zbMATH DE Number: 2093422
zbMATH Keywords: semi-invariants of quivers
zbMATH Keywords: cofree representations
zbMATH Keywords: modules of covariants
zbMATH Keywords: complete intersections
zbMATH Keywords: Dynkin type quivers
MaRDI profile type: Publication
full work available at URL: j.jpaa.2004.02.015
OpenAlex ID: W2027077491
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DOI: 10.1016/J.JPAA.2004.02.015
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description: scientific article; zbMATH DE number 2093422