The spectral radius of the Coxeter transformations for a generalized Cartan matrix
type: Item
instance of: scholarly article
title: The spectral radius of the Coxeter transformations for a generalized Cartan matrix
zbMATH Open document ID: 0819.15008
DOI: 10.1007/BF01450490
author: Claus Michael Ringel
published in: Mathematische Annalen
publication date: 1994-12-21T00:00:00Z
full work available at URL: https://eudml.org/doc/165256
review text: This work concerns the eigenvalues (more precisely, the spectral radius) of the Coxeter transformations \(C = C(A, \pi)\) for a generalized Cartan matrix \(A\) [for such matrices see \textit{V. Kac} ``Infinite dimensional Lie algebras'', 2nd ed. Cambridge Univ. Press (1985; Zbl 0574.17010); 3rd ed. (1990; Zbl 0716.17022)]. The following theorem is proved: Let \(A\) be a generalized Cartan matrix which is connected and neither of finite nor of affine type. Let \(C\) be a Coxeter transformation for \(A\). Then the spectral radius \(\rho (C) > 1\), and \(\rho (C)\) is an eigenvalue of multiplicity one, whereas any other eigenvalue \(\lambda\) of \(C\) satisfies \(| \lambda | < \rho (C)\).
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 17B67
Mathematics Subject Classification ID: 16G20
zbMATH DE Number: 704909
zbMATH Keywords: quivers
zbMATH Keywords: grip
zbMATH Keywords: eigenvalues
zbMATH Keywords: spectral radius
zbMATH Keywords: Coxeter transformations
zbMATH Keywords: Cartan matrix
reviewed by: Jan Kubarski
MaRDI profile type: Publication
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OpenAlex ID: W1998333671
description: scientific article; zbMATH DE number 704909