Spectral properties of Coxeter transformations and applications
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 \(\lambda^+_ X>0\), for an indecomposable preinjective or regular module X. We give some applications of our results.
scientific article; zbMATH DE number 4125612
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