Spectral properties of Coxeter transformations and applications
type: Item
instance of: scholarly article
title: Spectral properties of Coxeter transformations and applications
zbMATH Open document ID: 0687.16017
DOI: 10.1007/BF01189130
author: K. Appert
published in: Archiv der Mathematik
publication date: 1990-01-01T00:00:00Z
review text: Let k be an algebraically closed field. Let \(\Delta\) be a finite connected oriented graph without cycles. The path algebra \(A=k[\Delta]\) [see \textit{P. Gabriel}, in representation theory I, Lect. Notes Math. 831, 1-71 (1980; Zbl 0445.16023)] is a finite dimensional hereditary k- algebra. Let \(\Gamma_ A\) be the Auslander-Reiten quiver of A with translation \(\tau\), \(\Gamma_ A\) consists of a preprojective component \({\mathcal P}\), a set of regular components \({\mathcal R}\) and a preinjective component \({\mathcal I}.\) The representation-finite case (that is, when \(\Gamma_ A={\mathcal P}={\mathcal I}\) is finite) is well understood. In the tame case, linear methods where used to distinguish the preprojective, regular and preinjective modules [\textit{V. Dlab} and \textit{C. M. Ringel}: Indecomposable representations of graphs and algebras (Mem. Am. Math. Soc. 173, 1976; Zbl 0332.16015)]. In this paper we consider the wild case: let \(A=k[\Delta]\) and assume that \(\Delta\) is a bipartite graph which is neither of Dynkin nor Euclidean type. Let C be the Cartan matrix and \(\Phi =-C^{-t}C\) be the Coxeter matrix associated to A. Let \(\rho\) be the spectral radius of \(\Phi\) and \(y^+\) (resp. \(y^-)\) be a positive eigenvector of \(\Phi\) with eigenvalue \(\rho\) (resp. \(\rho^{-1})\). Then the linear function \(\delta^-,\delta^+: K_ 0(A)\to {\mathbb{Z}}\) with \(\delta^-(x)=y^- C^{-t}x^ t\), \(\delta^+(x)=xC^{-t}y^+\), characterize the position of an indecompoable module in the following way: \(X\in {\mathcal P}\) iff \(<y^-,\underline{\dim} X><0\); \(X\in {\mathcal R}\) iff \(<y^- ,\underline{\dim} X>>0\) and \(<\underline{\dim} X,y^+>>0\) and \(X\in {\mathcal I}\) iff \(<\underline{\dim} X,y^+><0\). Our second main result describes the asymptotic behaviour of (\underbar{dim} \(\tau^ mX)_{m\in {\mathbb{Z}}}\) for an indecomposable A-module X. Namely, we show that \[ \lim_{m\to \infty}(\dim \tau^ mX)/\rho^ m=\lambda^+_ Xy^+ \] with \(\lambda^+_ X>0\),…
Mathematics Subject Classification ID: 16P10
Mathematics Subject Classification ID: 16Gxx
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 15A21
Mathematics Subject Classification ID: 15A63
zbMATH DE Number: 4125612
zbMATH Keywords: wild algebras
zbMATH Keywords: finite connected oriented graph
zbMATH Keywords: path algebra
zbMATH Keywords: finite dimensional hereditary k-algebra
zbMATH Keywords: Auslander-Reiten quiver
zbMATH Keywords: translation
zbMATH Keywords: preprojective component
zbMATH Keywords: regular components
zbMATH Keywords: preinjective component
zbMATH Keywords: wild case
zbMATH Keywords: Cartan matrix
zbMATH Keywords: Coxeter matrix
zbMATH Keywords: eigenvalue
zbMATH Keywords: indecomposable module
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01189130
OpenAlex ID: W2055120715
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description: scientific article; zbMATH DE number 4125612