On the Coxeter transformation of a wild algebra
Let \(\Delta\) be a quiver which is not of Dynkin type, and \(B\) be a concealed algebra of type \(\Delta\). Denote by \(\phi_B\) the Coxeter matrix of \(B\), and by \(\rho_B\) the spectral radius of \(\phi_B\). The author shows that \(\rho_B\) is an eigenvalue of \(\phi_B\) with geometric multiplicity one, and that there is an eigenvector \(y^+\) of \(\phi_B\) (and an eigenvector \(y^-\) of \(\phi^{-1}_B\)) with all coordinates positive such that \(y^+\phi_B=\rho_By^+\) (and \(y^-\phi^{-1}_B=\rho_By^-\), respectively). She then applies this result to study the asymptotic behaviour of the (dimension vectors of the) indecomposable \(B\)-modules, obtaining generalisations of her results with \textit{J. A. de la Peña} [Arch. Math. 55, No. 2, 120-134 (1990; Zbl 0687.16017)]. As the author points out, recent results of \textit{C. M. Ringel} [Math. Ann. 300, No. 2, 331-339 (1994; Zbl 0819.15008)] (obtained after the submission of this paper) imply the main result, and can be used to obtain easier proofs for the others.
scientific article; zbMATH DE number 687053