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Spectral properties of Coxeter transformations and applications. (English) Zbl 0687.16017
Let k be an algebraically closed field. Let \(\Delta\) be a finite connected oriented graph without cycles. The path algebra \(A=k[\Delta]\) [see 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 [V. Dlab and 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 (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.
Reviewer: J.A.de la Peña

16P10 Finite rings and finite-dimensional associative algebras
16Gxx Representation theory of associative rings and algebras
15A18 Eigenvalues, singular values, and eigenvectors
15A21 Canonical forms, reductions, classification
15A63 Quadratic and bilinear forms, inner products
Full Text: DOI
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