Advertisement

A Generalization of Lefschetz Elements

  • Tadahito Harima
  • Toshiaki Maeno
  • Hideaki Morita
  • Yasuhide Numata
  • Akihito Wachi
  • Junzo Watanabe
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2080)

Abstract

In this chapter we would like to discuss a generalization of Lefschetz elements for an Artinian local ring to study the Jordan decomposition of a general element. The point of departure for us is Theorem 5.1 due to D. Rees. Several results from  Chap. 6 (e.g., stable ideals, Borel fixed ideals, gin(I), etc) are needed at a few points in  Chap. 5.

References

  1. 1.
    Ahn, J., Cho, Y.H., Park, J.P.: Generic initial ideals of Artinian ideals having Lefschetz properties or the strong Stanley property. J. Algebra 318(2), 589–606 (2007). doi:10.1016/j.jalgebra.2007.09.016. http://dx.doi.org/10.1016/j.jalgebra.2007.09.016 Google Scholar
  2. 2.
    Aigner, M.: Combinatorial theory. In: Classics in Mathematics. Springer, Berlin (1997). Reprint of the 1979 originalGoogle Scholar
  3. 3.
    Anderson, I.: Combinatorics of finite sets. Dover Publications Inc., Mineola (2002). Corrected reprint of the 1989 editionGoogle Scholar
  4. 4.
    Anderson, D.D., Winders, M.: Idealization of a module. J. Commut. Algebra 1(1), 3–56 (2009). doi:10.1216/JCA-2009-1-1-3. http://dx.doi.org/10.1216/JCA-2009-1-1-3 Google Scholar
  5. 5.
    Bass, H.: On the ubiquity of Gorenstein rings. Math. Z. 82, 8–28 (1963)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Bernšteĭn, I.N., Gelfand, I.M., Gelfand, S.I.: Schubert cells, and the cohomology of the spaces G ∕ P. Uspehi Mat. Nauk 28(3(171)), 3–26 (1973)Google Scholar
  7. 7.
    Boij, M., Laksov, D.: Nonunimodality of graded Gorenstein Artin algebras. Proc. Am. Math. Soc. 120(4), 1083–1092 (1994). doi:10.2307/2160222. http://dx.doi.org/10.2307/2160222 Google Scholar
  8. 8.
    Boij, M., Migliore, J.C., Miró-Roig, R.M., Nagel, U., Zanello, F.: On the shape of a pure O-sequence. Mem. Am. Math. Soc. 218(1024), viii + 78 (2012)Google Scholar
  9. 9.
    Bollobás, B.: Combinatorics: Set Systems, Hypergraphs, Families of Vectors and Combinatorial Probability. Cambridge University Press, Cambridge (1986).zbMATHGoogle Scholar
  10. 10.
    Brenner, H., Kaid, A.: Syzygy bundles on \({\mathbb{P}}^{2}\) and the weak Lefschetz property. Ill. J. Math. 51(4), 1299–1308 (2007). http://projecteuclid.org/getRecord?id=euclid.ijm/1258138545
  11. 11.
    de Bruijn, N.G., van Ebbenhorst Tengbergen, C., Kruyswijk, D.: On the set of divisors of a number. Nieuw Arch. Wiskunde (2) 23, 191–193 (1951)Google Scholar
  12. 12.
    Bruns, W., Herzog, J.: Cohen-Macaulay rings. In: Cambridge Studies in Advanced Mathematics, vol. 39. Cambridge University Press, Cambridge (1993)Google Scholar
  13. 13.
    Buchsbaum, D.A., Eisenbud, D.: Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension 3. Am. J. Math. 99(3), 447–485 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Canfield, E.R.: On a problem of Rota. Adv. Math. 29(1), 1–10 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Chen, C.P., Guo, A., Jin, X., Liu, G.: Trivariate monomial complete intersections and plane partitions. J. Commut. Algebra 3(4), 459–489 (2011). doi:10.1216/JCA-2011-3-4-459. http://dx.doi.org/10.1216/JCA-2011-3-4-459 Google Scholar
  16. 16.
    Chevalley, C.: Invariants of finite groups generated by reflections. Am. J. Math. 77, 778–782 (1955)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Chevalley, C.: Classification des groups de Lie algébriques. Séminaire C. Chevalley, 1956–1958, 2 vols. Secrétariat mathématique, Paris (1958)Google Scholar
  18. 18.
    Chevalley, C.: Sur les décompositions cellulaires des espaces G ∕ B. In: Algebraic Groups and Their Generalizations: Classical Methods, University Park, PA, 1991. Proceedings of the Symposium on Pure Mathematics, vol. 56, pp. 1–23. American Mathematical Society, Providence (1994). With a foreword by Armand BorelGoogle Scholar
  19. 19.
    Cho, Y.H., Park, J.P.: Conditions for generic initial ideals to be almost reverse lexicographic. J. Algebra 319(7), 2761–2771 (2008). doi:10.1016/j.jalgebra.2008.01.014. http://dx.doi.org/10.1016/j.jalgebra.2008.01.014 Google Scholar
  20. 20.
    Cimpoeaş, M.: Generic initial ideal for complete intersections of embedding dimension three with strong Lefschetz property. Bull. Math. Soc. Sci. Math. Roum. (N.S.) 50(98)(1), 33–66 (2007)Google Scholar
  21. 21.
    Cook II, D., Nagel, U.: The weak lefschetz property, monomial ideals, and lozenges. Illinois J. Math. 55(1), 377–395 (2012). MR3006693Google Scholar
  22. 22.
    Cook II, D., Nagel, U.: Enumerations deciding the weak lefschetz property (2011). Preprint, arXiv:1105.6062v2 [math.AC]Google Scholar
  23. 23.
    Conca, A.: Reduction numbers and initial ideals. Proc. Am. Math. Soc. 131(4), 1015–1020 (electronic) (2003). doi:10.1090/S0002-9939-02-06607-8. http://dx.doi.org/10.1090/S0002-9939-02-06607-8
  24. 24.
    Conca, A., Krattenthaler, C., Watanabe, J.: Regular sequences of symmetric polynomials. Rend. Semin. Mat. Univ. Padova 121, 179–199 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Constantinescu, A.: Hilbert function and Betti numbers of algebras with Lefschetz property of order m. Commun. Algebra 36(12), 4704–4720 (2008). doi:10.1080/00927870802174074. http://dx.doi.org/10.1080/00927870802174074 Google Scholar
  26. 26.
    Cook II, D., Nagel, U.: Hyperplane sections and the subtlety of the Lefschetz properties. J. Pure Appl. Algebra 216(1), 108–114 (2012). doi:10.1016/j.jpaa.2011.05.007. http://dx.doi.org/10.1016/j.jpaa.2011.05.007
  27. 27.
    Cox, D., Little, J., O’Shea, D.: Ideals, varieties, and algorithms: An introduction to computational algebraic geometry and commutative algebra. In: Undergraduate Texts in Mathematics, 3rd edn. Springer, New York (2007). doi:10.1007/978-0-387-35651-8. http://dx.doi.org/10.1007/978-0-387-35651-8.
  28. 28.
    Danilov, V.I.: The geometry of toric varieties. Uspekhi Mat. Nauk 33(2(200)), 85–134, 247 (1978)Google Scholar
  29. 29.
    Dilworth, R.P.: A decomposition theorem for partially ordered sets. Ann. Math. (2) 51, 161–166 (1950)Google Scholar
  30. 30.
    Dilworth, R.P.: Some combinatorial problems on partially ordered sets. In: Proceedings of Symposia in Applied Mathematics, vol. 10, pp. 85–90. American Mathematical Society, Providence (1960)Google Scholar
  31. 31.
    Eisenbud, D.: Commutative algebra. In: Graduate Texts in Mathematics, vol. 150. Springer, New York (1995). With a view toward algebraic geometryGoogle Scholar
  32. 32.
    Eliahou, S., Kervaire, M.: Minimal resolutions of some monomial ideals. J. Algebra 129(1), 1–25 (1990). doi:10.1016/0021-8693(90)90237-I. http://dx.doi.org/10.1016/0021-8693(90)90237-I Google Scholar
  33. 33.
    Engel, K.: Sperner theory. In: Encyclopedia of Mathematics and Its Applications, vol. 65. Cambridge University Press, Cambridge (1997). doi:10.1017/CBO9780511574719. http://dx.doi.org/10.1017/CBO9780511574719
  34. 34.
    Engel, K., Gronau, H.D.O.F.: Sperner theory in partially ordered sets. In: Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], vol. 78. BSB B.G. Teubner Verlagsgesellschaft, Leipzig (1985). With German, French and Russian summariesGoogle Scholar
  35. 35.
    Erdős, P.: Extremal problems in number theory. In: Proceedings of Symposia in Pure Mathematics, vol. VIII, pp. 181–189. American Mathematical Society, Providence (1965)Google Scholar
  36. 36.
    Freese, R.: An application of Dilworth’s lattice of maximal antichains. Discrete Math. 7, 107–109 (1974)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Fulton, W.: Young tableaux. In: London Mathematical Society Student Texts, vol. 35. Cambridge University Press, Cambridge (1997). With applications to representation theory and geometryGoogle Scholar
  38. 38.
    Geramita, A.V.: Inverse systems of fat points: Waring’s problem, secant varieties of Veronese varieties and parameter spaces for Gorenstein ideals. In: The Curves Seminar at Queen’s, vol. X (Kingston, ON, 1995). Queen’s Papers in Pure and Applied Mathematics, vol. 102, pp. 2–114. Queen’s University, Kingston (1996)Google Scholar
  39. 39.
    Geramita, A.V., Harima, T., Migliore, J.C., Shin, Y.S.: The Hilbert function of a level algebra. Mem. Am. Math. Soc. 186(872), vi + 139 (2007)Google Scholar
  40. 40.
    Goodman, R., Wallach, N.R.: Representations and invariants of the classical groups. In: Encyclopedia of Mathematics and its Applications, vol. 68. Cambridge University Press, Cambridge (1998)Google Scholar
  41. 41.
    Gordan, P., Nöther, M.: Ueber die algebraischen Formen, deren Hesse’sche Determinante identisch verschwindet. Math. Ann. 10(4), 547–568 (1876). doi:10.1007/BF01442264. http://dx.doi.org/10.1007/BF01442264 Google Scholar
  42. 42.
    Greene, C., Kleitman, D.J.: Proof techniques in the theory of finite sets. In: Studies in Combinatorics. MAA Studies in Mathematics, vol. 17, pp. 22–79. Mathematical Association of America, Washington (1978)Google Scholar
  43. 43.
    Griffiths, P., Harris, J.: Principles of Algebraic Geometry. Wiley Classics Library. Wiley, New York (1994). Reprint of the 1978 originalGoogle Scholar
  44. 44.
    Gröbner, W.: Über irreduzible Ideale in kommutativen Ringen. Math. Ann. 110(1), 197–222 (1935). doi:10.1007/BF01448025. http://dx.doi.org/10.1007/BF01448025
  45. 45.
    Gunston, T.K.: Cohomological degrees, Dilworth numbers and linear resolution. Thesis (Ph.D.)–Rutgers The State University of New Jersey - New Brunswick. ProQuest LLC, Ann Arbor (1998). http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:9915442
  46. 46.
    Hall, P.: On representatives of subsets. J. Lond. Math. Soc. S1-10(1), 26–30 (1935). doi:10.1112/jlms/s1-10.37.26Google Scholar
  47. 47.
    Hall, B.C.: Lie groups, Lie algebras, and representations. In: Graduate Texts in Mathematics, vol. 222. Springer, New York (2003). An elementary introductionGoogle Scholar
  48. 48.
    Hara, M., Watanabe, J.: The determinants of certain matrices arising from the Boolean lattice. Discrete Math. 308(23), 5815–5822 (2008). doi:10.1016/j.disc.2007.09.055. http://dx.doi.org/10.1016/j.disc.2007.09.055
  49. 49.
    Harima, T., Wachi, A.: Generic initial ideals, graded Betti numbers, and k-Lefschetz properties. Commun. Algebra 37(11), 4012–4025 (2009). doi:10.1080/00927870802502753. http://dx.doi.org/10.1080/00927870802502753 Google Scholar
  50. 50.
    Harima, T., Watanabe, J.: The finite free extension of Artinian K-algebras with the strong Lefschetz property. Rend. Sem. Mat. Univ. Padova 110, 119–146 (2003). See errata in [51].Google Scholar
  51. 51.
    Harima, T., Watanabe, J.: Erratum to: “The finite free extension of Artinian K-algebras with the strong Lefschetz property” [Rend. Sem. Mat. Univ. Padova 110, 119–146 (2003); mr2033004]. Rend. Sem. Mat. Univ. Padova 112, 237–238 (2004)MathSciNetzbMATHGoogle Scholar
  52. 52.
    Harima, T., Watanabe, J.: The central simple modules of Artinian Gorenstein algebras. J. Pure Appl. Algebra 210(2), 447–463 (2007). doi:10.1016/j.jpaa.2006.10.016. http://dx.doi.org/10.1016/j.jpaa.2006.10.016 Google Scholar
  53. 53.
    Harima, T., Watanabe, J.: The strong Lefschetz property for Artinian algebras with non-standard grading. J. Algebra 311(2), 511–537 (2007). doi:10.1016/j.jalgebra.2007.01.019. http://dx.doi.org/10.1016/j.jalgebra.2007.01.019 Google Scholar
  54. 54.
    Harima, T., Watanabe, J.: The commutator algebra of a nilpotent matrix and an application to the theory of commutative Artinian algebras. J. Algebra 319(6), 2545–2570 (2008). doi:10.1016/j.jalgebra.2007.09.011. http://dx.doi.org/10.1016/j.jalgebra.2007.09.011
  55. 55.
    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). doi:10.1016/S0021-8693(03)00038-3. http://dx.doi.org/10.1016/S0021-8693(03)00038-3 Google Scholar
  56. 56.
    Harima, T., Sakaki, S., Wachi, A.: Generic initial ideals of some monomial complete intersections in four variables. Arch. Math. (Basel) 94(2), 129–137 (2010). doi:10.1007/s00013-009-0088-2. http://dx.doi.org/10.1007/s00013-009-0088-2 Google Scholar
  57. 57.
    Herzog, J., Popescu, D.: The strong lefschetz property and simple extensions (2005). Preprint, arXiv:math/0506537Google Scholar
  58. 58.
    Hiller, H.L.: Schubert calculus of a Coxeter group. Enseign. Math. (2) 27(1, 2), 57–84 (1981)Google Scholar
  59. 59.
    Hiller, H.: Geometry of Coxeter groups. In: Research Notes in Mathematics, vol. 54. Pitman (Advanced Publishing Program), Boston (1982)Google Scholar
  60. 60.
    Humphreys, J.E.: Introduction to Lie algebras and representation theory. In: Graduate Texts in Mathematics, vol. 9. Springer, New York (1978). Second printing, revisedGoogle Scholar
  61. 61.
    Humphreys, J.E.: Reflection groups and Coxeter groups. In: Cambridge Studies in Advanced Mathematics, vol. 29. Cambridge University Press, Cambridge (1990)Google Scholar
  62. 62.
    Huneke, C., Ulrich, B.: General hyperplane sections of algebraic varieties. J. Algebr. Geom. 2(3), 487–505 (1993)MathSciNetzbMATHGoogle Scholar
  63. 63.
    Huybrechts, D., Lehn, M.: The Geometry of Moduli Spaces of Sheaves, 2nd edn. Cambridge Mathematical Library. Cambridge University Press, Cambridge (2010). doi:10.1017/CBO9780511711985. http://dx.doi.org/10.1017/CBO9780511711985
  64. 64.
    Iarrobino, A.: Associated graded algebra of a Gorenstein Artin algebra. Mem. Am. Math. Soc. 107(514), viii + 115 (1994)Google Scholar
  65. 65.
    Iarrobino, A., Kanev, V.: Power sums, Gorenstein algebras, and determinantal loci. In: Lecture Notes in Mathematics, vol. 1721. Springer, Berlin (1999). Appendix C by Iarrobino and Steven L. KleimanGoogle Scholar
  66. 66.
    Ikeda, H.: Results on Dilworth and Rees numbers of Artinian local rings. Jpn. J. Math. (N.S.) 22(1), 147–158 (1996)Google Scholar
  67. 67.
    Ikeda, H., Watanabe, J.: The Dilworth lattice of Artinian rings. J. Commut. Algebra 1(2), 315–326 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  68. 68.
    Jacobson, N.: Lie Algebras. Dover Publications, New York (1979). Republication of the 1962 originalGoogle Scholar
  69. 69.
    Jurkiewicz, J.: Chow ring of projective nonsingular torus embedding. Colloq. Math. 43(2), 261–270 (1980/1981)Google Scholar
  70. 70.
    Kantor, W.M.: On incidence matrices of finite projective and affine spaces. Math. Z. 124, 315–318 (1972)MathSciNetzbMATHCrossRefGoogle Scholar
  71. 71.
    Kaveh, K.: Note on cohomology rings of spherical varieties and volume polynomial. J. Lie Theory 21(2), 263–283 (2011)MathSciNetzbMATHGoogle Scholar
  72. 72.
    Kleiman, S.L.: Toward a numerical theory of ampleness. Ann. Math. (2) 84, 293–344 (1966)Google Scholar
  73. 73.
    Koszul, J.L.: Homologie et Cohomologie des Algébre de Lie. Bull. Soc. Math. de France 78, 65–127 (1950)MathSciNetzbMATHGoogle Scholar
  74. 74.
    Krattenthaler, C.: Advanced determinant calculus. Sém. Lothar. Combin. 42, Art. B42q, 67 pp. (electronic) (1999). The Andrews Festschrift (Maratea, 1998)Google Scholar
  75. 75.
    Krattenthaler, C.: Another involution principle-free bijective proof of Stanley’s hook-content formula. J. Comb. Theory Ser. A 88(1), 66–92 (1999). doi:10.1006/jcta.1999.2979. http://dx.doi.org/10.1006/jcta.1999.2979 Google Scholar
  76. 76.
    Krull, W.: Idealtheorie. Zweite, ergänzte Auflage. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 46. Springer, Berlin (1968)Google Scholar
  77. 77.
    Kuhnigk, K.: On Macaulay duals of Hilbert ideals. J. Pure Appl. Algebra 210(2), 473–480 (2007). doi:10.1016/j.jpaa.2006.10.014. http://dx.doi.org/10.1016/j.jpaa.2006.10.014 Google Scholar
  78. 78.
    Lazarsfeld, R.: Positivity in algebraic geometry, I. Classical setting: line bundles and linear series. In: Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 48. Springer, Berlin (2004).Google Scholar
  79. 79.
    Lefschetz, S.: L’analysis situs et la géométrie algébrique. Gauthier-Villars, Paris (1950)zbMATHGoogle Scholar
  80. 80.
    Li, J., Zanello, F.: Monomial complete intersections, the weak Lefschetz property and plane partitions. Discrete Math. 310(24), 3558–3570 (2010). doi:10.1016/j.disc.2010.09.006. http://dx.doi.org/10.1016/j.disc.2010.09.006
  81. 81.
    Lindsey, M.: A class of Hilbert series and the strong Lefschetz property. Proc. Am. Math. Soc. 139(1), 79–92 (2011). doi:10.1090/S0002-9939-2010-10498-7. http://dx.doi.org/10.1090/S0002-9939-2010-10498-7
  82. 82.
    Lossen, C.: When does the Hessian determinant vanish identically? (On Gordan and Noether’s proof of Hesse’s claim). Bull. Braz. Math. Soc. (N.S.) 35(1), 71–82 (2004). doi:10.1007/s00574-004-0004-0. http://dx.doi.org/10.1007/s00574-004-0004-0
  83. 83.
    Macaulay, F.S.: On the resolution of a given modular system into primary systems including some properties of Hilbert numbers. Math. Ann. 74(1), 66–121 (1913). doi:10.1007/BF01455345. http://dx.doi.org/10.1007/BF01455345 Google Scholar
  84. 84.
    Macaulay, F.S.: The Algebraic Theory of Modular Systems. Cambridge Mathematical Library. Cambridge University Press, Cambridge (1994). Revised reprint of the 1916 original, With an introduction by Paul RobertsGoogle Scholar
  85. 85.
    Macdonald, I.G.: Symmetric Functions and Hall Polynomials, 2nd edn. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York (1995). With contributions by A. Zelevinsky, Oxford Science PublicationsGoogle Scholar
  86. 86.
    MacMahon, P.A.: Combinatory Analysis. Two volumes (bound as one). Chelsea Publishing, New York (1960)Google Scholar
  87. 87.
    Maeno, T.: Lefschetz property, Schur-Weyl duality and a q-deformation of Specht polynomial. Commun. Algebra 35(4), 1307–1321 (2007). doi:10.1080/00927870601142371. http://dx.doi.org/10.1080/00927870601142371 Google Scholar
  88. 88.
    Maeno, T., Numata, Y.: Sperner property and finite-dimensional gorenstein algebras associated to matroids (2011). Preprint, arXiv:1107.5094Google Scholar
  89. 89.
    Maeno, T., Watanabe, J.: Lefschetz elements of Artinian Gorenstein algebras and Hessians of homogeneous polynomials. Ill. J. Math. 53(2), 591–603 (2009). http://projecteuclid.org/getRecord?id=euclid.ijm/1266934795 Google Scholar
  90. 90.
    Maeno, T., Numata, Y., Wachi, A.: Strong Lefschetz elements of the coinvariant rings of finite Coxeter groups. Algebras Represent. Theory 14(4), 625–638 (2011). doi:10.1007/s10468-010-9207-9. http://dx.doi.org/10.1007/s10468-010-9207-9
  91. 91.
    Martsinkovsky, A., Vlassov, A.: The representation rings of k[x]. preprint (2004). http://www.math.neu.edu/~martsinkovsky/GreenExcerpt.pdf
  92. 92.
    Matlis, E.: Injective modules over Noetherian rings. Pac. J. Math. 8, 511–528 (1958)MathSciNetzbMATHCrossRefGoogle Scholar
  93. 93.
    Matsumura, H.: Commutative ring theory. In: Cambridge Studies in Advanced Mathematics, vol. 8, 2nd edn. Cambridge University Press, Cambridge (1989). Translated from the Japanese by M. ReidGoogle Scholar
  94. 94.
    McDaniel, C.: The strong lefschetz property for coinvariant rings of finite reflection groups. J. Algebra 331, 68–95 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  95. 95.
    McMullen, P.: The maximum numbers of faces of a convex polytope. Mathematika 17, 179–184 (1970)MathSciNetzbMATHCrossRefGoogle Scholar
  96. 96.
    McMullen, P.: The numbers of faces of simplicial polytopes. Isr. J. Math. 9, 559–570 (1971)MathSciNetzbMATHCrossRefGoogle Scholar
  97. 97.
    Meyer, D.M., Smith, L.: The Lasker-Noether theorem for unstable modules over the Steenrod algebra. Commun. Algebra 31(12), 5841–5845 (2003). doi:10.1081/AGB-120024856. http://dx.doi.org/10.1081/AGB-120024856 Google Scholar
  98. 98.
    Meyer, D.M., Smith, L.: Realization and nonrealization of Poincaré duality quotients of \(\mathbb{F}_{2}[x,y]\) as topological spaces. Fund. Math. 177(3), 241–250 (2003). doi:10.4064/fm177-3-4. http://dx.doi.org/10.4064/fm177-3-4
  99. 99.
    Meyer, D.M., Smith, L.: Poincaré duality algebras, Macaulay’s dual systems, and Steenrod operations. In: Cambridge Tracts in Mathematics, vol. 167. Cambridge University Press, Cambridge (2005). doi:10.1017/CBO9780511542855. http://dx.doi.org/10.1017/CBO9780511542855
  100. 100.
    Mezzetti, E., Miró-Roig, R.M., Ottaviani, G.: Laplace equations and the weak lefschetz property. Can. J. Math. (2012, online first). doi:10.4153/CJM-2012-033-x. http://dx.doi.org/10.4153/CJM-2012-033-x
  101. 101.
    Migliore, J., Nagel, U.: A tour of the weak and strong lefschetz properties (2011). Preprint, arXiv:1109.5718v2 [math.AC]Google Scholar
  102. 102.
    Migliore, J.C., Miró-Roig, R.M., Nagel, U.: Monomial ideals, almost complete intersections and the weak Lefschetz property. Trans. Am. Math. Soc. 363(1), 229–257 (2011). doi:10.1090/S0002-9947-2010-05127-X. http://dx.doi.org/10.1090/S0002-9947-2010-05127-X Google Scholar
  103. 103.
    Morita, H., Wachi, A., Watanabe, J.: Zero-dimensional Gorenstein algebras with the action of the symmetric group. Rend. Semin. Mat. Univ. Padova 121, 45–71 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  104. 104.
    Motzkin, T.S.: Comonotone curves and polyhedra. In: Bull. Am. Math. Soc. [154], p. 35. doi:10.1090/S0002-9904-1957-10068-8. http://dx.doi.org/10.1090/S0002-9904-1957-10068-8
  105. 105.
    Mumford, D.: Stability of projective varieties. L’Enseignement Mathématique, Geneva (1977). Lectures given at the “Institut des Hautes Études Scientifiques”, Bures-sur-Yvette, March-April 1976, Monographie de l’Enseignement Mathématique, No. 24Google Scholar
  106. 106.
    Murthy, M.P.: A note on factorial rings. Arch. Math. (Basel) 15, 418–420 (1964)Google Scholar
  107. 107.
    Nagata, M.: Local rings. In: Interscience Tracts in Pure and Applied Mathematics, vol. 13. Interscience Publishers a division of Wiley, New York (1962)Google Scholar
  108. 108.
    Numata, Y., Wachi, A.: The strong Lefschetz property of the coinvariant ring of the Coxeter group of type H 4. J. Algebra 318(2), 1032–1038 (2007). doi:10.1016/j.jalgebra.2007.06.016. http://dx.doi.org/10.1016/j.jalgebra.2007.06.016
  109. 109.
    Oda, T.: Torus embeddings and applications. In: Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 57. Tata Institute of Fundamental Research, Bombay (1978). Based on joint work with Katsuya MiyakeGoogle Scholar
  110. 110.
    Oda, T.: Convex bodies and algebraic geometry. In: Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 15. Springer, Berlin (1988). An introduction to the theory of toric varieties, Translated from the JapaneseGoogle Scholar
  111. 111.
    Okon, J.S., Vicknair, J.P.: A Gorenstein ring with larger Dilworth number than Sperner number. Can. Math. Bull. 43(1), 100–104 (2000). doi:10.4153/CMB-2000-015-2. http://dx.doi.org/10.4153/CMB-2000-015-2
  112. 112.
    Okon, J.S., Rush, D.E., Vicknair, J.P.: Numbers of generators of ideals in a group ring of an elementary abelian p-group. J. Algebra 224(1), 1–22 (2000). doi:10.1006/jabr.1999.8082. http://dx.doi.org/10.1006/jabr.1999.8082 Google Scholar
  113. 113.
    Okonek, C., Schneider, M., Spindler, H.: Vector bundles on complex projective spaces. In: Progress in Mathematics, vol. 3. Birkhäuser Boston, Boston (1980)Google Scholar
  114. 114.
    Pandharipande, R.: A compactification over \(\overline{M}_{g}\) of the universal moduli space of slope-semistable vector bundles. J. Am. Math. Soc. 9(2), 425–471 (1996). doi:10.1090/S0894-0347-96-00173-7. http://dx.doi.org/10.1090/S0894-0347-96-00173-7
  115. 115.
    Perles, M.A.: A proof of Dilworth’s decomposition theorem for partially ordered sets. Isr. J. Math. 1, 105–107 (1963)MathSciNetzbMATHCrossRefGoogle Scholar
  116. 116.
    Popov, V.L.: On the stability of the action of an algebraic group on an algebraic variety. Math. USSR Izvestiya 6(2), 367 (1972). doi:doi:10.1070/IM1972v006n02ABEH001877. http://dx.doi.org/10.1070/IM1972v006n02ABEH001877
  117. 117.
    Proctor, R.A.: Solution of two difficult combinatorial problems with linear algebra. Am. Math. Mon. 89(10), 721–734 (1982). doi:10.2307/2975833. http://dx.doi.org/10.2307/2975833 Google Scholar
  118. 118.
    Pukhlikov, A.V., Khovanskiĭ, A.G.: The Riemann-Roch theorem for integrals and sums of quasipolynomials on virtual polytopes. Algebra i Analiz 4(4), 188–216 (1992)Google Scholar
  119. 119.
    Reid, L., Roberts, L.G., Roitman, M.: On complete intersections and their Hilbert functions. Can. Math. Bull. 34(4), 525–535 (1991). doi:10.4153/CMB-1991-083-9. http://dx.doi.org/10.4153/CMB-1991-083-9
  120. 120.
    Roberts, P.C.: A computation of local cohomology. In: Commutative Algebra: Syzygies, Multiplicities, and Birational Algebra, South Hadley, MA, 1992. Contemporary Mathematics, vol. 159, pp. 351–356. American Mathematical Society, Providence (1994)Google Scholar
  121. 121.
    Sagan, B.E.: The symmetric group: Representations, combinatorial algorithms, and symmetric functions. In: Graduate Texts in Mathematics, vol. 203, 2nd edn. Springer, New York (2001).Google Scholar
  122. 122.
    Sally, J.D.: Numbers of generators of ideals in local rings. Marcel Dekker, New York (1978)zbMATHGoogle Scholar
  123. 123.
    Sekiguchi, H.: The upper bound of the Dilworth number and the Rees number of Noetherian local rings with a Hilbert function. Adv. Math. 124(2), 197–206 (1996). doi:10.1006/aima.1996.0082. http://dx.doi.org/10.1006/aima.1996.0082
  124. 124.
    Shephard, G.C., Todd, J.A.: Finite unitary reflection groups. Can. J. Math. 6, 274–304 (1954)MathSciNetzbMATHCrossRefGoogle Scholar
  125. 125.
    Shioda, T.: On the graded ring of invariants of binary octavics. Am. J. Math. 89, 1022–1046 (1967)MathSciNetzbMATHCrossRefGoogle Scholar
  126. 126.
    Smith, L.: Note on the realization of complete intersections algebras as the cohomology of a space. Quart. J. Math. Oxford Ser 33, 379–384 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  127. 127.
    Smith, L.: Polynomial invariants of finite groups. In: Research Notes in Mathematics, vol. 6. A K Peters Ltd., Wellesley (1995)Google Scholar
  128. 128.
    Solomon, L.: Invariants of finite reflection groups. Nagoya Math. J. 22, 57–64 (1963)MathSciNetzbMATHGoogle Scholar
  129. 129.
    Solomon, L.: Partition identities and invariants of finite reflection groups. Nagoya Math. J. 22, 57–64 (1963)MathSciNetzbMATHGoogle Scholar
  130. 130.
    Solomon, L.: Invariants of euclidian reflection groups. Trans. Am. Math. Soc. 113, 274–286 (1964)zbMATHCrossRefGoogle Scholar
  131. 131.
    Specht, W.: Die irreduziblen Darstellungen der symmetrischen Gruppe. Math. Z. 39(1), 696–711 (1935). doi:10.1007/BF01201387. http://dx.doi.org/10.1007/BF01201387
  132. 132.
    Sperner, E.: Ein Satz über Untermengen einer endlichen Menge. Math. Z. 27(1), 544–548 (1928). doi:10.1007/BF01171114. http://dx.doi.org/10.1007/BF01171114
  133. 133.
    Stanley, R.P.: Cohen-Macaulay complexes. In: Higher Combinatorics, Proc. NATO Advanced Study Inst., Berlin, 1976, pp. 51–62. NATO Adv. Study Inst. Ser., Ser. C: Math. and Phys. Sci., 31. Reidel, Dordrecht (1977)Google Scholar
  134. 134.
    Stanley, R.P.: Hilbert functions of graded algebras. Adv. Math. 28(1), 57–83 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  135. 135.
    Stanley, R.P.: The number of faces of a simplicial convex polytope. Adv. Math. 35(3), 236–238 (1980). doi:10.1016/0001-8708(80)90050-X. http://dx.doi.org/10.1016/0001-8708(80)90050-X Google Scholar
  136. 136.
    Stanley, R.P.: Weyl groups, the hard Lefschetz theorem, and the Sperner property. SIAM J. Algebr. Discrete Methods 1(2), 168–184 (1980). doi:10.1137/0601021. http://dx.doi.org/10.1137/0601021
  137. 137.
    Stanley, R.P.: Combinatorics and commutative algebra. In: Progress in Mathematics, vol. 41, 2nd edn. Birkhäuser Boston, Boston (1996)Google Scholar
  138. 138.
    Stanley, R.P.: Enumerative combinatorics, vol. 1. In: Cambridge Studies in Advanced Mathematics, vol. 49. Cambridge University Press, Cambridge (1997). With a foreword by Gian-Carlo Rota, Corrected reprint of the 1986 originalGoogle Scholar
  139. 139.
    Steinberg, R.: Differential equations invariant under finite reflection groups. Trans. Am. Math. Soc. 112, 392–400 (1964)zbMATHCrossRefGoogle Scholar
  140. 140.
    Stong, R.E.: Poincaré algebras modulo an odd prime. Comment. Math. Helv. 49, 382–407 (1974)MathSciNetzbMATHCrossRefGoogle Scholar
  141. 141.
    Stong, R.E.: Cup products in Grassmannians. Topol. Appl. 13(1), 103–113 (1982). doi:10.1016/0166-8641(82)90012-8. http://dx.doi.org/10.1016/0166-8641(82)90012-8
  142. 142.
    Terasoma, T., Yamada, H.: Higher Specht polynomials for the symmetric group. Proc. Jpn. Acad. Ser. A Math. Sci. 69(2), 41–44 (1993). http://projecteuclid.org/getRecord?id=euclid.pja/1195511538
  143. 143.
    Trung, N.V.: Bounds for the minimum numbers of generators of generalized Cohen-Macaulay ideals. J. Algebra 90(1), 1–9 (1984). doi:10.1016/0021-8693(84)90193-5. http://dx.doi.org/10.1016/0021-8693(84)90193-5
  144. 144.
    Watanabe, J.: A note on Gorenstein rings of embedding codimension three. Nagoya Math. J. 50, 227–232 (1973)MathSciNetzbMATHGoogle Scholar
  145. 145.
    Watanabe, J.: Some remarks on Cohen-Macaulay rings with many zero divisors and an application. J. Algebra 39(1), 1–14 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  146. 146.
    Watanabe, J.: The Dilworth number of Artinian rings and finite posets with rank function. In: Commutative Algebra and Combinatorics, Kyoto, 1985. Advanced Studies in Pure Mathematics, vol. 11, pp. 303–312. North-Holland, Amsterdam (1987)Google Scholar
  147. 147.
    Watanabe, J.: \(\mathfrak{m}\)-full ideals. Nagoya Math. J. 106, 101–111 (1987). http://projecteuclid.org/getRecord?id=euclid.nmj/1118780704
  148. 148.
    Watanabe, J.: A note on complete intersections of height three. Proc. Am. Math. Soc. 126(11), 3161–3168 (1998). doi:10.1090/S0002-9939-98-04477-3. http://dx.doi.org/10.1090/S0002-9939-98-04477-3
  149. 149.
    Watanabe, J.: A remark on the hessian of homogeneous polynomials. In: Bogoyavlenskij, O., Coleman, A.J., Geramita, A.V., Ribenboim, P. (eds.) The Curves Seminar at Queen’s, vol. XIII. Queen’s Papers in Pure and Applied Mathematics, vol. 119, pp. 171–178. Queen’s University, Kingston (2000)Google Scholar
  150. 150.
    Watanabe, J.: On the minimal number of the quotient of a complete intersection by a regular sequence. Proc. Sch. Sci. Tokai Univ. 47, 1–10 (2012)MathSciNetzbMATHGoogle Scholar
  151. 151.
    Watanabe, J.: On the theory of Gordan-Noether on the homogeneous forms with zero Hessian. Proc. Sch. Tokai Univ. 48 (2013) (to appear)Google Scholar
  152. 152.
    Weyl, H.: The classical groups. In: Princeton Landmarks in Mathematics. Princeton University Press, Princeton (1997). Their invariants and representations, Fifteenth printing, Princeton PaperbacksGoogle Scholar
  153. 153.
    Wiebe, A.: The Lefschetz property for componentwise linear ideals and Gotzmann ideals. Commun. Algebra 32(12), 4601–4611 (2004). doi:10.1081/AGB-200036809. http://dx.doi.org/10.1081/AGB-200036809 Google Scholar
  154. 154.
    Youngs, J.W.T.: The November meeting in Evanston. Bull. Am. Math. Soc. 63(1), 29–38 (1957). doi:10.1090/S0002-9904-1957-10068-8. http://dx.doi.org/10.1090/S0002-9904-1957-10068-8

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Tadahito Harima
    • 1
  • Toshiaki Maeno
    • 2
  • Hideaki Morita
    • 3
  • Yasuhide Numata
    • 4
  • Akihito Wachi
    • 5
  • Junzo Watanabe
    • 6
  1. 1.Department of Mathematics EducationNiigata UniversityNiigataJapan
  2. 2.Department of MathematicsMeijo UniversityNagoyaJapan
  3. 3.Muroran Institute of TechnologyMuroranJapan
  4. 4.Department of Mathematical SciencesShinshu UniversityMatsumotoJapan
  5. 5.Department of MathematicsHokkaido University of EducationKushiroJapan
  6. 6.Department of MathematicsTokai UniversityHiratsukaJapan

Personalised recommendations