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Properties of the Coxeter transformations for the affine Dynkin cycle

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Abstract

We study spectral properties of the Coxeter transformations for the affine Dynkin cycle and find the Jordan form of the Coxeter transformation and the Coxeter numbers.

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References

  1. Bourbaki, N. Lie Groups and Lie Algebras. Coxeter Groups and Tits Systems. Groups Generated by Reflections. Root Systems (Mir, Moscow, 1972) [Russian translation].

    MATH  Google Scholar 

  2. Bernstein, I. N., Gel’fand, I. M., and Ponomarev, V. A. “Coxeter Functors, and Gabriel’s Theorem,” Usp. Mat. Nauk 28, No. 2, 19–33 (1973).

    MATH  MathSciNet  Google Scholar 

  3. Subbotin, V. F. and Stekol’shchik, R. B. “Jordan Form of the Coxeter Transformation and Applications to Representations of Finite Graphs,” Funktsional. Anal. i Prilozhen. 12, No. 4, 97 (1978).

    MATH  MathSciNet  Google Scholar 

  4. Stekolshchik, R. B. “Coxeter Numbers of Extended Dynkin Diagrams with Multiple Bonds,” in Proceedings of XVI All-Union Algebraic Conference. Part 2 (Leningrad, 1981), P. 187.

    Google Scholar 

  5. Kolmykov V. A. and Men’shikh, V. V. “On Switching of Graph Orientations,” Discrete Math. Appl. 12, No. 5, 453–457 (2002).

    MATH  MathSciNet  Google Scholar 

  6. Men’shikh, V. V. and Kolmykov, V. A. “Invariant Transformations of Signed Graphs,” Avtomatika i Vychislitel’naya Tekhnika 34, No. 5, 51–57 (2000).

    Google Scholar 

  7. Menshikh, V. V. and Subbotin, V. F. “Graph Orientations and the Generated Coxeter Transformations,” Available from VINITI, No. 6479-82 (Voronezh Univ., 1982).

    Google Scholar 

  8. Coleman, A. J. “Killing and the Coxeter Transformation of Kac-Moody Algebras,” Inventiones mathematicae 95, No. 3, 447–477 (1989).

    Article  MATH  MathSciNet  Google Scholar 

  9. Drozd, Ju. A. “Coxeter Transformations and Representations of Partially Ordered Sets,” Funkcional. Anal. i Prilozen. 8, No. 3, 34–42 (1974).

    MathSciNet  Google Scholar 

  10. Stekol’shchik, R. B. “Regularity Conditions for Graph Representations,” Trudy Matem. Fakul’teta VGU, No. 16, 58–61 (1975).

    Google Scholar 

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Authors and Affiliations

  1. Voronezh Institute of Ministry of Internal Affairs of Russia, pr. Patriotov 53, Voronezh, 394065, Russia

    V. V. Men’shikh

  2. Voronezh State University, Universitetskaya pl. 1, Voronezh, 394006, Russia

    V. F. Subbotin

Authors
  1. V. V. Men’shikh
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  2. V. F. Subbotin
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Correspondence to V. V. Men’shikh.

Additional information

Original Russian Text © V.V. Men’shikh, V.F. Subbotin, 2014, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2014, No. 9, pp. 43–48.

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Men’shikh, V.V., Subbotin, V.F. Properties of the Coxeter transformations for the affine Dynkin cycle. Russ Math. 58, 36–40 (2014). https://doi.org/10.3103/S1066369X14090047

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  • DOI: https://doi.org/10.3103/S1066369X14090047

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