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Description of posets critical with respect to the nonnegativity of the quadratic Tits form

  • V. M. Bondarenko
  • M. V. Stepochkina
Article

We present the complete description of finite posets whose Tits form is not nonnegative but all proper subsets of which have nonnegative Tits forms. A similar result for positive forms was obtained by the authors earlier.

References

  1. 1.
    P. Gabriel, “Unzerlegbare Darstellungen,” Manuscr. Math., 6, 71–103, 309 (1972).CrossRefMathSciNetGoogle Scholar
  2. 2.
    S. Brenner, “Quivers with commutativity conditions and some phenomenology of forms,” in: Proc. of the Internat. Conf. on Representations of Algebras, Carleton University, Ottawa (1974), Paper No. 5.Google Scholar
  3. 3.
    Yu. A. Drozd, “Coxeter transformations and representations of posets,” Funkts. Anal. Prilozhen., 8, 34–42 (1974).MathSciNetGoogle Scholar
  4. 4.
    M. M. Kleiner and A. V. Roiter, “Representations of differential graded categories,” in: Matrix Problems [in Russian], Institute of Mathematics, Academy of Sciences of the Ukrainian SSR (1977), pp. 5–70.Google Scholar
  5. 5.
    L. A. Nazarova and A. V. Roiter, “Representations of posets,” Zap. Nauch. Semin. LOMI, 28, 5–31 (1972).MathSciNetGoogle Scholar
  6. 6.
    V. M. Bondarenko and A. M. Polishchuk, “On the criterion of positive definiteness for one class of infinite quadratic forms,” Nelin. Kolyv., 6, No. 1, 3–14 (2003).MathSciNetGoogle Scholar
  7. 7.
    V. M. Bondarenko, “On (min, max)-equivalence of posets and applications to the Tits forms,” Visn. Kyiv Univ., Ser. Fiz. Mat., No. 1, 24–25 (2005).Google Scholar
  8. 8.
    V. M. Bondarenko and M. V. Styopochkina, “On posets of width two with positive Tits form,” Algebra Discrete Math., No. 2, 11–22 (2005).Google Scholar
  9. 9.
    V. M. Bondarenko and M. V. Stepochkina, “Posets of the injective-finite type,” Nauk. Visn. Uzhhorod Univ., Ser. Mat. Inform., Issue 9, 15–25 (2005).Google Scholar
  10. 10.
    V. M. Bondarenko and M. V. Stepochkina, “(Min, max)-equivalence of posets and the quadratic Tits form,” in: Collection of Works “Problems of Analysis and Algebra,” Institute of Mathematics, Ukrainian National Academy of Sciences, 2, No. 3 (2005), pp. 18–58.Google Scholar
  11. 11.
    V. M. Bondarenko and M. V. Styopochkina, “On finite posets of inj-finite type and their Tits forms,” Algebra Discrete Math., No. 2, 17–21 (2006).Google Scholar
  12. 12.
    V. M. Bondarenko and M. V. St’opochkina, “On the serial posets with positive-definite quadratic Tits form,” Nelin. Kolyv., 9, No. 3, 320–325 (2006).MathSciNetGoogle Scholar
  13. 13.
    V. M. Bondarenko and M. V. St’opochkina, “On the form of posets with positive-definite Tits form,” Visn. Kyiv Univ., Issue 3, 11–14 (2006).Google Scholar
  14. 14.
    V. M. Bondarenko and M. V. St’opochkina, “On the relationship between the inj-finiteness of the type and positive-definiteness of the quadratic Tits form for finite posets,” Nauk. Visn. Uzhhorod Univ., Issue 12, 20–27 ( 2006).Google Scholar
  15. 15.
    V. M. Bondarenko and M. V. Stepochkina, “(Min, max)-equivalence of posets and nonnegative Tits forms,” Ukr. Mat. Zh., 60, No. 9, 1157–1167 (2008).CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2009

Authors and Affiliations

  • V. M. Bondarenko
    • 1
  • M. V. Stepochkina
    • 1
  1. 1.Institute of MathematicsUkrainian National Academy of SciencesKievUkraine

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