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The spectral radius of the Coxeter transformations for a generalized Cartan matrix. (English) Zbl 0819.15008
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 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 $$| \lambda | < \rho (C)$$.

##### MSC:
 15A18 Eigenvalues, singular values, and eigenvectors 17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras 16G20 Representations of quivers and partially ordered sets
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##### References:
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