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.
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References
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
Aigner, M.: Combinatorial theory. In: Classics in Mathematics. Springer, Berlin (1997). Reprint of the 1979 original
Anderson, I.: Combinatorics of finite sets. Dover Publications Inc., Mineola (2002). Corrected reprint of the 1989 edition
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
Bass, H.: On the ubiquity of Gorenstein rings. Math. Z. 82, 8–28 (1963)
Bernšteĭn, I.N., Gel′fand, I.M., Gel′fand, S.I.: Schubert cells, and the cohomology of the spaces G ∕ P. Uspehi Mat. Nauk 28(3(171)), 3–26 (1973)
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
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)
Bollobás, B.: Combinatorics: Set Systems, Hypergraphs, Families of Vectors and Combinatorial Probability. Cambridge University Press, Cambridge (1986).
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
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)
Bruns, W., Herzog, J.: Cohen-Macaulay rings. In: Cambridge Studies in Advanced Mathematics, vol. 39. Cambridge University Press, Cambridge (1993)
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)
Canfield, E.R.: On a problem of Rota. Adv. Math. 29(1), 1–10 (1978)
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
Chevalley, C.: Invariants of finite groups generated by reflections. Am. J. Math. 77, 778–782 (1955)
Chevalley, C.: Classification des groups de Lie algébriques. Séminaire C. Chevalley, 1956–1958, 2 vols. Secrétariat mathématique, Paris (1958)
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 Borel
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
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)
Cook II, D., Nagel, U.: The weak lefschetz property, monomial ideals, and lozenges. Illinois J. Math. 55(1), 377–395 (2012). MR3006693
Cook II, D., Nagel, U.: Enumerations deciding the weak lefschetz property (2011). Preprint, arXiv:1105.6062v2 [math.AC]
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
Conca, A., Krattenthaler, C., Watanabe, J.: Regular sequences of symmetric polynomials. Rend. Semin. Mat. Univ. Padova 121, 179–199 (2009)
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
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
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.
Danilov, V.I.: The geometry of toric varieties. Uspekhi Mat. Nauk 33(2(200)), 85–134, 247 (1978)
Dilworth, R.P.: A decomposition theorem for partially ordered sets. Ann. Math. (2) 51, 161–166 (1950)
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)
Eisenbud, D.: Commutative algebra. In: Graduate Texts in Mathematics, vol. 150. Springer, New York (1995). With a view toward algebraic geometry
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
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
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 summaries
Erdős, P.: Extremal problems in number theory. In: Proceedings of Symposia in Pure Mathematics, vol. VIII, pp. 181–189. American Mathematical Society, Providence (1965)
Freese, R.: An application of Dilworth’s lattice of maximal antichains. Discrete Math. 7, 107–109 (1974)
Fulton, W.: Young tableaux. In: London Mathematical Society Student Texts, vol. 35. Cambridge University Press, Cambridge (1997). With applications to representation theory and geometry
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)
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)
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)
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
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)
Griffiths, P., Harris, J.: Principles of Algebraic Geometry. Wiley Classics Library. Wiley, New York (1994). Reprint of the 1978 original
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
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
Hall, P.: On representatives of subsets. J. Lond. Math. Soc. S1-10(1), 26–30 (1935). doi:10.1112/jlms/s1-10.37.26
Hall, B.C.: Lie groups, Lie algebras, and representations. In: Graduate Texts in Mathematics, vol. 222. Springer, New York (2003). An elementary introduction
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
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
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].
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)
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
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
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
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
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
Herzog, J., Popescu, D.: The strong lefschetz property and simple extensions (2005). Preprint, arXiv:math/0506537
Hiller, H.L.: Schubert calculus of a Coxeter group. Enseign. Math. (2) 27(1, 2), 57–84 (1981)
Hiller, H.: Geometry of Coxeter groups. In: Research Notes in Mathematics, vol. 54. Pitman (Advanced Publishing Program), Boston (1982)
Humphreys, J.E.: Introduction to Lie algebras and representation theory. In: Graduate Texts in Mathematics, vol. 9. Springer, New York (1978). Second printing, revised
Humphreys, J.E.: Reflection groups and Coxeter groups. In: Cambridge Studies in Advanced Mathematics, vol. 29. Cambridge University Press, Cambridge (1990)
Huneke, C., Ulrich, B.: General hyperplane sections of algebraic varieties. J. Algebr. Geom. 2(3), 487–505 (1993)
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
Iarrobino, A.: Associated graded algebra of a Gorenstein Artin algebra. Mem. Am. Math. Soc. 107(514), viii + 115 (1994)
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. Kleiman
Ikeda, H.: Results on Dilworth and Rees numbers of Artinian local rings. Jpn. J. Math. (N.S.) 22(1), 147–158 (1996)
Ikeda, H., Watanabe, J.: The Dilworth lattice of Artinian rings. J. Commut. Algebra 1(2), 315–326 (2009)
Jacobson, N.: Lie Algebras. Dover Publications, New York (1979). Republication of the 1962 original
Jurkiewicz, J.: Chow ring of projective nonsingular torus embedding. Colloq. Math. 43(2), 261–270 (1980/1981)
Kantor, W.M.: On incidence matrices of finite projective and affine spaces. Math. Z. 124, 315–318 (1972)
Kaveh, K.: Note on cohomology rings of spherical varieties and volume polynomial. J. Lie Theory 21(2), 263–283 (2011)
Kleiman, S.L.: Toward a numerical theory of ampleness. Ann. Math. (2) 84, 293–344 (1966)
Koszul, J.L.: Homologie et Cohomologie des Algébre de Lie. Bull. Soc. Math. de France 78, 65–127 (1950)
Krattenthaler, C.: Advanced determinant calculus. Sém. Lothar. Combin. 42, Art. B42q, 67 pp. (electronic) (1999). The Andrews Festschrift (Maratea, 1998)
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
Krull, W.: Idealtheorie. Zweite, ergänzte Auflage. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 46. Springer, Berlin (1968)
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
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).
Lefschetz, S.: L’analysis situs et la géométrie algébrique. Gauthier-Villars, Paris (1950)
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
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
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
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
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 Roberts
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 Publications
MacMahon, P.A.: Combinatory Analysis. Two volumes (bound as one). Chelsea Publishing, New York (1960)
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
Maeno, T., Numata, Y.: Sperner property and finite-dimensional gorenstein algebras associated to matroids (2011). Preprint, arXiv:1107.5094
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
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
Martsinkovsky, A., Vlassov, A.: The representation rings of k[x]. preprint (2004). http://www.math.neu.edu/~martsinkovsky/GreenExcerpt.pdf
Matlis, E.: Injective modules over Noetherian rings. Pac. J. Math. 8, 511–528 (1958)
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. Reid
McDaniel, C.: The strong lefschetz property for coinvariant rings of finite reflection groups. J. Algebra 331, 68–95 (2011)
McMullen, P.: The maximum numbers of faces of a convex polytope. Mathematika 17, 179–184 (1970)
McMullen, P.: The numbers of faces of simplicial polytopes. Isr. J. Math. 9, 559–570 (1971)
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
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
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
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
Migliore, J., Nagel, U.: A tour of the weak and strong lefschetz properties (2011). Preprint, arXiv:1109.5718v2 [math.AC]
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
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)
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
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. 24
Murthy, M.P.: A note on factorial rings. Arch. Math. (Basel) 15, 418–420 (1964)
Nagata, M.: Local rings. In: Interscience Tracts in Pure and Applied Mathematics, vol. 13. Interscience Publishers a division of Wiley, New York (1962)
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
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 Miyake
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 Japanese
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
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
Okonek, C., Schneider, M., Spindler, H.: Vector bundles on complex projective spaces. In: Progress in Mathematics, vol. 3. Birkhäuser Boston, Boston (1980)
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
Perles, M.A.: A proof of Dilworth’s decomposition theorem for partially ordered sets. Isr. J. Math. 1, 105–107 (1963)
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
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
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)
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
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)
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).
Sally, J.D.: Numbers of generators of ideals in local rings. Marcel Dekker, New York (1978)
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
Shephard, G.C., Todd, J.A.: Finite unitary reflection groups. Can. J. Math. 6, 274–304 (1954)
Shioda, T.: On the graded ring of invariants of binary octavics. Am. J. Math. 89, 1022–1046 (1967)
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)
Smith, L.: Polynomial invariants of finite groups. In: Research Notes in Mathematics, vol. 6. A K Peters Ltd., Wellesley (1995)
Solomon, L.: Invariants of finite reflection groups. Nagoya Math. J. 22, 57–64 (1963)
Solomon, L.: Partition identities and invariants of finite reflection groups. Nagoya Math. J. 22, 57–64 (1963)
Solomon, L.: Invariants of euclidian reflection groups. Trans. Am. Math. Soc. 113, 274–286 (1964)
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
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
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)
Stanley, R.P.: Hilbert functions of graded algebras. Adv. Math. 28(1), 57–83 (1978)
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
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
Stanley, R.P.: Combinatorics and commutative algebra. In: Progress in Mathematics, vol. 41, 2nd edn. Birkhäuser Boston, Boston (1996)
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 original
Steinberg, R.: Differential equations invariant under finite reflection groups. Trans. Am. Math. Soc. 112, 392–400 (1964)
Stong, R.E.: Poincaré algebras modulo an odd prime. Comment. Math. Helv. 49, 382–407 (1974)
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
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
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
Watanabe, J.: A note on Gorenstein rings of embedding codimension three. Nagoya Math. J. 50, 227–232 (1973)
Watanabe, J.: Some remarks on Cohen-Macaulay rings with many zero divisors and an application. J. Algebra 39(1), 1–14 (1976)
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)
Watanabe, J.: \(\mathfrak{m}\)-full ideals. Nagoya Math. J. 106, 101–111 (1987). http://projecteuclid.org/getRecord?id=euclid.nmj/1118780704
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
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)
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)
Watanabe, J.: On the theory of Gordan-Noether on the homogeneous forms with zero Hessian. Proc. Sch. Tokai Univ. 48 (2013) (to appear)
Weyl, H.: The classical groups. In: Princeton Landmarks in Mathematics. Princeton University Press, Princeton (1997). Their invariants and representations, Fifteenth printing, Princeton Paperbacks
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
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
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Harima, T., Maeno, T., Morita, H., Numata, Y., Wachi, A., Watanabe, J. (2013). A Generalization of Lefschetz Elements. In: The Lefschetz Properties. Lecture Notes in Mathematics, vol 2080. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38206-2_5
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