Mathematische Zeitschrift

, Volume 264, Issue 4, pp 813–828 | Cite as

A primitive derivation and logarithmic differential forms of Coxeter arrangements

Article

Abstract

Let W be a finite irreducible real reflection group, which is a Coxeter group. We explicitly construct a basis for the module of differential 1-forms with logarithmic poles along the Coxeter arrangement by using a primitive derivation. As a consequence, we extend the Hodge filtration, indexed by nonnegative integers, into a filtration indexed by all integers. This filtration coincides with the filtration by the order of poles. The results are translated into the derivation case.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abe, T.: A generalized logarithmic module and duality of Coxeter multiarrangements. arXiv.0807.2552v1Google Scholar
  2. 2.
    Abe, T., Yoshinaga, M.: Coxeter multiarrangements with quasi-constant multiplicities. arXiv:0708.3228Google Scholar
  3. 3.
    Bourbaki, N.: Groupes et Algèbres de Lie. Chapitres 4,5 et 6, Hermann, Paris (1968)Google Scholar
  4. 4.
    Dubrovin, B.: Geometry of 2D topological field theories. In: Francaviglia, M., Greco, S. (ed.) Integrable Systems and Quantum Groups. Lectures at C.I.M.E., 1993, LNM, vol. 1620, pp. 120–348. Springer, Berlin (1996)Google Scholar
  5. 5.
    Orlik, P., Terao, H.: Arrangements of Hyperplanes. Grundlehren der Mathematischen Wissenschaften, vol. 300. Springer, Berlin (1992)Google Scholar
  6. 6.
    Saito, K.: On the uniformization of complements of discriminant loci. In: Conference Notes. Am. Math. Soc. Summer Institute, Williamstown (1975)Google Scholar
  7. 7.
    Saito K.: On a linear structure of the quotient variety by a finite reflection group. Publ. RIMS Kyoto Univ. 29, 535–579 (1993)MATHCrossRefGoogle Scholar
  8. 8.
    Saito, K.: Uniformization of the orbifold of a finite reflection Group. RIMS 1414. preprint (2003)Google Scholar
  9. 9.
    Solomon L.: Invariants of finite reflection groups. Nagoya Math. J. 22, 57–64 (1963)MATHMathSciNetGoogle Scholar
  10. 10.
    Solomon L., Terao H.: The double Coxeter arrangements. Comment. Math. Helv. 73, 237–258 (1998)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Terao H.: Multiderivations of Coxeter arrangements. Invent. Math. 148, 659–674 (2002)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Terao H.: Bases of the contact-order filtration of derivations of Coxeter arrangements. Proc. Am. Math. Soc. 133, 2029–2034 (2005)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Terao H.: The Hodge filtration and the contact-order filtration of derivations of Coxeter arrangements. Manuscr. Math. 118, 1–9 (2005)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Terao H.: A correction to Bases of the contact-order filtration of derivations of Coxeter arrangements. Proc. Amer. Math. Soc. 136, 2639 (2008)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Yoshinaga M.: The primitive derivation and freeness of multi-Coxeter arrangements. Proc. Jpn. Acad. Ser. A 78(7), 116–119 (2002)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Ziegler, G.M.: Multiarrangements of hyperplanes and their freeness. In: Singularities (Iowa City, IA, 1986), Contemp. Math., vol. 90, pp. 345–359. Amer. Math. Soc., Providence (1989)Google Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Department of MathematicsKyoto UniversityKyotoJapan
  2. 2.Department of MathematicsHokkaido UniversitySapporo, HokkaidoJapan

Personalised recommendations