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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

##### MSC:
 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
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