Conditions on a finite number of orbits for \(A_r\)-type quivers.
type: Item
instance of: scholarly article
title: Conditions on a finite number of orbits for \(A_r\)-type quivers.
zbMATH Open document ID: 1080.16009
author: Makoto Nagura
author: Tsuyoshi Niitani
published in: Journal of Algebra
publication date: 2004-10-01T00:00:00Z
review text: Let \(Q\) be a quiver of type \(A_r\). The objective of the paper is the variety \(R_d\) of \(\mathbb{C}\)-representations of \(Q\) of fixed dimension vector \(d=(d_1,\dots,d_r)\), equipped with the natural action of a group \(G\) which has the form \(\prod_{i=1}^rG_i\), where each \(G_i\) is either \(\text{Gl}(d_i,\mathbb{C})\) or \(\text{Sl}(d_i,\mathbb{C})\). \(\mathbb{C}\) denotes the field of complex numbers. It is well known that when \(G_i=\text{Gl}(d_i,\mathbb{C})\) for every \(i\) then this action has finitely many orbits. One of the main results of the paper characterizes precisely those dimension vectors and groups \(G\) of the above form which admit only a finite number of orbits in \(R_d\). The condition does not depend on the orientation of \(Q\). Moreover, an explicit description of all orbits in \(R_d\) is given in case their number is finite. (Also submitted to MR.)
reviewed by: Stanisław Kasjan
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 16G70
zbMATH DE Number: 2105356
zbMATH Keywords: representations of quivers
zbMATH Keywords: varieties of representations
zbMATH Keywords: algebraic group actions
zbMATH Keywords: numbers of orbits
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2003.11.021
OpenAlex ID: W2067367008
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DOI: 10.1016/J.JALGEBRA.2003.11.021
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description: scientific article; zbMATH DE number 2105356