Polynomial invariants of representations of quivers
type: Item
instance of: scholarly article
title: Polynomial invariants of representations of quivers
zbMATH Open document ID: 0816.16015
DOI: 10.1007/BF02564477
author: Stephen Donkin
published in: Commentarii Mathematici Helvetici
publication date: 1995-07-13T00:00:00Z
full work available at URL: https://eudml.org/doc/140329
review text: For a quiver \(Q\) with \(n\) vertices, a given dimension vector \(\alpha = (\alpha_ 1,\dots, \alpha_ n)\), and an algebraically closed field \(k\), let \(E_ 1,\dots, E_ n\) be fixed \(k\)-vector spaces with \(\dim_ k E_ i = \alpha_ i\). The space \(R(Q,\alpha)\) of \(k\)-representations of \(Q\) is the product \(\prod \text{Hom}_ k(E_ i, E_ j)\) over the arrows \(i \to j\) in \(Q\). The group \(\text{GL}(\alpha) = \text{GL}(E_ 1) \times \dots \times\text{GL}(E_ n)\) naturally acts on \(R(Q,\alpha)\) and thus on the symmetric algebra \(k[R(Q,\alpha)]\). For \(\text{char }k = 0\), \textit{L. Le Bruyn} and \textit{C. Procesi} [Trans. Am. Math. Soc. 317, 585- 598 (1990; Zbl 0693.16018)] have determined a complete system of polynomial invariants of \(R(Q,\alpha)\), namely the coefficients of characteristic polynomials of products of maps along oriented cycles in \(Q\). The author obtains this result for arbitrary characteristic by use of his previous paper on ``Invariants of several matrices'' [Invent. Math. 110, 389-401 (1992)]. He points out that the result remains valid over \(\mathbb{Z}\) with stability under base change. Moreover, he proves a generalized statement with \(\text{GL}(\alpha)\) replaced by a subgroup \(\prod\text{GL}(E_ i)_{\theta_ i}\), where \(\text{GL}(E_ i)_{\theta_ i}\) denotes the centralizer of some element \(\theta_ i \in \text{End}_ k(E_ i)\).
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 14L30
Mathematics Subject Classification ID: 20G05
zbMATH DE Number: 590901
zbMATH Keywords: quivers
zbMATH Keywords: \(k\)-representations
zbMATH Keywords: symmetric algebras
zbMATH Keywords: polynomial invariants
zbMATH Keywords: characteristic polynomials
zbMATH Keywords: base change
reviewed by: Wolfgang Rump
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 590901