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, Volume 118, Issue 1, pp 1–9 | Cite as

The Hodge filtration and the contact-order filtration of derivations of Coxeter arrangements



The Hodge filtration of the module of derivations on the orbit space of a finite real reflection group acting on an ℓ-dimensional Euclidean space was introduced and studied by K. Saito [5] [6]. It is closely related to the flat structure or the Frobenius manifold structure. We will show that the Hodge filtration coincides with the filtration by the order of contacts to the reflecting hyperplanes. Moreover, a standard basis for the Hodge filtration is explicitly given.

Mathematics Subject Classification (2000)



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  1. 1.
    Bourbaki, N.: Groupes et Algèbres de Lie. Chapitres 4, 5 et 6, Hermann, Paris 1968Google Scholar
  2. 2.
    Dubrovin, B.: Geometry of 2D topological field theories. In: Francaviglia. M., Greco, S (eds), ‘‘Integrable systems and quantum groups’’ Lectures at C.I.M.E., 1993, LNM 1620, Springer, Berlin-Heidelberg-New York, 1996, pp 120–348Google Scholar
  3. 3.
    Edelman, P., Reiner, V.: Free arrangements and rhombic tilings. Discrete and Computational Geometry 15, 307–340 (1996)Google Scholar
  4. 4.
    Orlik, P., Terao, H.: Arrangements of Hyperplanes. Grundlehren der Math. Wiss. 300, Springer Verlag, 1992Google Scholar
  5. 5.
    Saito, K.: On a linear structure of a quotient variety by a finite reflexion group. RIMS Kyoto preprint 288, 1979 = Publ. Res. Inst. Math. Sci. 29, 535–579 (1993)Google Scholar
  6. 6.
    Saito, K.: Finite reflection groups and related geometry (A motivation to the period mapping for primitive forms). Preprint, 2000Google Scholar
  7. 7.
    Solomon, L., Terao, H.: The double Coxeter arrangement. Comment. Math. Helv. 73, 237–258 (1998)CrossRefGoogle Scholar
  8. 8.
    Terao, H.: Multiderivations of Coxeter arrangements. Inventiones math. 148, 659–674 (2002)CrossRefGoogle Scholar
  9. 9.
    Yoshinaga, M.: The primitive derivation and freeness of multi-Coxeter arrangements. Proc. Japan Acad. Ser. A Math. Sci. 78, 116–119 (2002)Google Scholar
  10. 10.
    Yoshinaga, M.: Some characterizations of freeness of hyperplane arrangement. Inventiones math. 157, 449–454 (2004)CrossRefGoogle Scholar
  11. 11.
    Ziegler, G. M.: Multiarrangements of hyperplanes and their freeness. In: Singularities. Contemporary Math. 90, Amer. Math. Soc., 1989, pp 345–359Google Scholar

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© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Mathematics DepartmentTokyo Metropolitan UniversityTokyoJapan

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