×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] A’Campo, N.: Sur les valeurs propres de la transformation de Coxeter. Invent. Math.33 (1976), 61-67 · Zbl 0406.20041 · doi:10.1007/BF01425505
[2] Berman, S.; Lee, Y.S.; Moody, R.V.: The spectrum of a Coxeter transformation, affine Coxeter transformations, and the defect map. J. Algebra121 (1989), 339-357 · Zbl 0679.17007 · doi:10.1016/0021-8693(89)90070-7
[3] Berman, S.; Moody, R.V.; Wonenburger, M.: Cartan matrices with null roots and finite Cartan matrices. Indiana Math. J.21 (1972), 1091-1099 · Zbl 0245.17005 · doi:10.1512/iumj.1972.21.21087
[4] Bernstein, I.N.; Gelfand, I.M.: Ponomarev, V.A.: Coxeter functors and Gabriel’s theorem. Usp. Mat. Nauk. Russ. Math. Surv.28 (1973), 17-32 · Zbl 0279.08001 · doi:10.1070/RM1973v028n02ABEH001526
[5] Coleman, A.J.: Killing and the Coxeter transformation of Kac-Moody algebras. Invent. Math.95 (1989), 447-477 · Zbl 0679.17008 · doi:10.1007/BF01393885
[6] Dlab, V.; Ringel, C.M.: Eigenvalues of Coxeter transformations and the Gelfand-Kirillov dimension of the preprojective algebras. Proc. Am. Math. Soc.83 (1981), 228-232 · Zbl 0471.15005
[7] Gantmacher, F.R.: Matrizentheorie II. Berlin (1966)
[8] Kac, V.: Infinite dimensional Lie algebras. 2nd ed. Cambridge University Press (1985) · Zbl 0574.17010
[9] de la Pe?a, J.; Takane, M.: Spectral properties of Coxeter transformations and applications. Arch. Math.55 (1990), 120-134 · Zbl 0687.16017 · doi:10.1007/BF01189130
[10] Seneta, E.: Non-negative matrices and Markov chains. Springer (1973) · Zbl 0278.15011
[11] Subbotin, V.F.; Stekol’shchik, R.B.: Jordan form of Coxeter transformations and applications to representations of finite graphs. Funk. Anal. Prilo?.12 (1978); Engl. transl. Funct. Anal. Appl.12 (1978), 67-68
[12] Takane, M.: On the Coxeter transformation of a wild algebra. (to appear) · Zbl 0841.16016
[13] Zhang, Y.: Eigenvalues of Coxeter transformations and the structure of the regular components of the Auslander-Reiten quiver. Commun. Algebra.17 (1989), 2347-2362 · Zbl 0686.16025 · doi:10.1080/00927878908823853
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.