Algebras and Representation Theory

, Volume 14, Issue 4, pp 625–638 | Cite as

Strong Lefschetz Elements of the Coinvariant Rings of Finite Coxeter Groups

  • Toshiaki Maeno
  • Yasuhide Numata
  • Akihito WachiEmail author


For the coinvariant rings of finite Coxeter groups of types other than H4, we show that a homogeneous element of degree one is a strong Lefschetz element if and only if it is not fixed by any reflections. We also give the necessary and sufficient condition for strong Lefschetz elements in the invariant subrings of the coinvariant rings of Weyl groups.


Coxeter group Weyl group Coinvariant ring Strong Lefschetz property Flag variety Hard Lefschetz theorem 

Mathematics Subject Classifications (2010)

Primary—20F55; Secondary—13A50 14M15 14N15 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bernšteǐn, I.N., Gel’fand, I.M., Gel’fand, S.I.: Schubert cells, and the cohomology of the spaces G/P. Usp. Mat. Nauk 28(3, 171), 3–26 (1973) (This article has appeared in English translation [Russ. Math. Surv. 28(3), 1–26 (1973)]) MR MR0429933 (55 #2941)Google Scholar
  2. 2.
    Borel, A.: Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts. Ann. Math. (2) 57, 115–207 (1953) MR MR0051508 (14,490e)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Chevalley, C.: Séminaire C. Chevalley, 1956–1958. Classification des groupes de Lie algébriques, 2 vols. Secrétariat Mathématique, 11 rue Pierre Curie, Paris (1958) MR MR0106966 (21 #5696)Google Scholar
  4. 4.
    Griffiths, P., Harris, J.: Principles of algebraic geometry. In: Pure and Applied Mathematics. Wiley-Interscience, Wiley & Sons, New York (1978) MR MR507725 (80b:14001)Google Scholar
  5. 5.
    Hiller, H.L.: Schubert calculus of a Coxeter group. Enseign. Math. (2) 27(1–2), 57–84 (1981) MR MR630960 (82m:14031)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Harima, T., Migliore, J.C., Nagel, U., Watanabe, J.: The weak and strong Lefschetz properties for Artinian K-algebras. J. Algebra 262(1), 99–126 (2003) MR MR1970804 (2004b:13001)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Humphreys, J.E.: Reflection groups and Coxeter groups. In: Cambridge Studies in Advanced Mathematics, vol. 29. Cambridge University Press, Cambridge (1990) MR MR1066460 (92h:20002)Google Scholar
  8. 8.
    Kleiman, S.L.: Toward a numerical theory of ampleness. Ann. Math. (2) 84, 293–344 (1966) MR MR0206009 (34 #5834)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Mehta, M.L.: Basic sets of invariant polynomials for finite reflection groups. Commun. Algebra 16(5), 1083–1098 (1988) MR MR926338 (88m:20104)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Numata, Y., Wachi, A.: The strong Lefschetz property of the coinvariant ring of the Coxeter group of type \(H\sb 4\). J. Algebra 318(2), 1032–1038 (2007) MR MR2371985zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Smith, L.: Polynomial invariants of finite groups. In: Research Notes in Mathematics, vol. 6. A K Peters Ltd., Wellesley, MA (1995) MR MR1328644 (96f:13008)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  • Toshiaki Maeno
    • 1
  • Yasuhide Numata
    • 2
    • 3
  • Akihito Wachi
    • 4
    Email author
  1. 1.Department of Electrical EngineeringKyoto UniversityKyotoJapan
  2. 2.Department of Mathematical InformaticsThe University of TokyoTokyoJapan
  3. 3.Japan Science and Technology Agency (JST), CRESTTokyoJapan
  4. 4.Department of MathematicsHokkaido University of EducationKushiroJapan

Personalised recommendations