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Hilbert functions of graded algebras. (English) Zbl 0384.13012


MSC:

13H15 Multiplicity theory and related topics
13E05 Commutative Noetherian rings and modules
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
14M15 Grassmannians, Schubert varieties, flag manifolds
13G05 Integral domains
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[1] Atiyah, M. F.; MacDonald, I. G., Introduction to Commutative Algebra (1969), Addison-Wesley: Addison-Wesley Reading, Mass. · Zbl 0175.03601
[2] Bennett, B., On the characteristic functions of a local ring, Ann. of Math., 91, 25-87 (1970) · Zbl 0198.06101
[3] Buchsbaum, D. A.; Eisenbud, D., Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension 3, Amer. J. Math., 99, 447-485 (1977) · Zbl 0373.13006
[4] Burnside, W., Theory of Groups of Finite Order (1911), Cambridge Univ. Press, reprinted by Dover, New York, 1955 · JFM 42.0151.02
[5] Clements, G.; Lindström, B., A generalization of a combinatorial theorem of Macaulay, J. Combinatorial Theory, 7, 230-238 (1969) · Zbl 0186.01704
[6] S. Goto; S. Goto
[7] S. Goto, N. Suzuki, and K. Watanabe; S. Goto, N. Suzuki, and K. Watanabe · Zbl 0361.20066
[8] C. Greene and D. J. Kleitmanin; C. Greene and D. J. Kleitmanin · Zbl 0409.05012
[9] Gröbner, W., Moderne Algebraische Geometrie (1949), Springer-Verlag: Springer-Verlag Vienna · Zbl 0033.12706
[10] Herzog, J.; Kunz, E., Die Wertehalbgruppe eines lokalen Rings der Dimension 1, Ber. Heidelberger Akad. Wiss., II (1971), 1971 · Zbl 0212.06102
[11] (Herzog, J.; Kunz, E., Der kanonische Modul eines Cohen-Macaulay-Rings. Der kanonische Modul eines Cohen-Macaulay-Rings, Lecture Notes in Mathematics, no. 238 (1971), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0231.13009
[12] Hironaka, H., Certain numerical characters of singularities, J. Math. Kyoto Univ., 10, 151-187 (1970) · Zbl 0214.20003
[13] Hochster, M.; Eagon, J. A., Cohen-Macaulay rings, invariant theory, and the generic kperfection of determimental loci, Amer. J. Math., 93, 1020-1058 (1971) · Zbl 0244.13012
[14] Hochster, M., Rings of invariants of tori, Cohen-Macaulay rings generated by monomials, and polytopes, Ann. of Math., 96, 318-337 (1972) · Zbl 0233.14010
[15] Hodge, W. D.; Pedoe, D., (Methods of Algebraic Geometry, Vol. II (1968), Cambridge Univ. Press: Cambridge Univ. Press London/New York) · Zbl 0157.27501
[16] Kaplansky, I., Commutative Rings (1974), Univ. of Chicago Press · Zbl 0203.34601
[17] Kleiman, S. L.; Laksov, D., Schubert calculus, Amer. Math. Monthly, 79, 1061-1082 (1972) · Zbl 0272.14016
[18] Macaulay, F. S., The Algebraic Theory of Modular Systems, (Cambridge Tracts in Mathematics and Mathematical Physics (1916), Cambridge Univ. Press: Cambridge Univ. Press London), No. 19 · Zbl 0802.13001
[19] Macaulay, F. S., Some properties of enumeration in the theory of modular systems, (Proc. London Math. Soc., 26 (1927)), 531-555 · JFM 53.0104.01
[20] MacMahon, P. A., (Combinatory Analysis, Vols. 1-2 (1916), Cambridge Univ. Press: Cambridge Univ. Press London), reprinted by Chelsea, New York, 1960 · JFM 46.0118.07
[21] Mallows, C. L.; Sloane, N. J.A, On the invariants of a linear group of order 336, (Proc. Cambridge Philos. Soc., 74 (1973)), 435-440 · Zbl 0268.20033
[22] McMullen, P., The number of faces of simplicial polytopes, Israel J. Math., 9, 559-570 (1971) · Zbl 0209.53701
[23] Molien, T., Über die Invarianten der Linearen Substitutionsgruppen, Sitz. König. Preuss. Akad. Wiss., 1152-1156 (1897) · JFM 28.0115.01
[24] Popoviciu, T., Studie şi cercetari ştiintifice, Acad. R.P.R. Filiala Cluj, 4, 8 (1953)
[25] Reiten, I., The converse to a theorem of Sharp on Gorenstein modules, (Proc. Amer. Math. Soc., 32 (1972)), 417-420 · Zbl 0235.13016
[26] Smoke, W., Dimension and multiplicity for graded algebras, J. Algebra, 21, 149-173 (1972) · Zbl 0231.13006
[27] Sperner, E., Über einen kombinatorischen Satz von Macaulay und seine Anwendung auf die Theorie der Polynomideale, Abh. Math. Sem. Univ. Hamburg, 7, 149-163 (1930) · JFM 55.0663.01
[28] Stanley, R., Theory and application of plane partitions, Parts 1 and 2, Stud. in Appl. Math., 50, 259-279 (1971) · Zbl 0225.05012
[29] Stanley, R., Ordered structures and partitions, Mem. Amer. Math. Soc., 119 (1972) · Zbl 0246.05007
[30] Stanley, R., Linear homogeneous diophantine equations and magic labelings of graphs, Duke Math. J., 40, 607-632 (1973) · Zbl 0269.05109
[31] Stanley, R., Combinatorial reciprocity theorems, Advances in Math., 14, 194-253 (1974) · Zbl 0294.05006
[32] Stanley, R., Combinatorial reciprocity theorems, (Hall, M.; Van Lint, J. H., Combinatorics. Combinatorics, Mathematical Centre Tracts (1974), Mathematisch Centrum: Mathematisch Centrum Amsterdam), 107-118, No. 56 · Zbl 0299.05008
[33] Stanley, R., Cohen-Macaulay rings and constructible polytopes, Bull. Amer. Math. Soc., 81, 133-135 (1975) · Zbl 0304.52005
[34] Stanley, R., The upper bound conjecture and Cohen-Macaulay rings, Stud. in Appl. Math., 54, 135-142 (1975) · Zbl 0308.52009
[35] Stanley, R., Some combinatorial aspects of the Schubert calculus, (Proc. Table Ronde, Combinatoire et Représentation du Groupe Symétrique. Proc. Table Ronde, Combinatoire et Représentation du Groupe Symétrique, Strasbourg (26-30 Avril 1976). Proc. Table Ronde, Combinatoire et Représentation du Groupe Symétrique. Proc. Table Ronde, Combinatoire et Représentation du Groupe Symétrique, Strasbourg (26-30 Avril 1976), Lecture Notes in Mathematics, no. 579 (1977), Springer: Springer Berlin), 217-251 · Zbl 0359.05006
[36] Svanes, T., Coherent cohomology of Schubert subschemes of flag schemes and applications, Advances in Math., 14, 369-453 (1974) · Zbl 0308.14008
[37] Watanabe, K., Certain invariant subrings are Gorenstein, I, Osaka J. Math., 11, 1-8 (1974) · Zbl 0281.13007
[38] Watanabe, K., Certain invariant subrings are Gorenstein, II, Osaka J. Math., 11, 379-388 (1974) · Zbl 0292.13008
[39] Whipple, F., On a theorem due to F. S. Macaulay, J. London Math. Soc., 8, 431-437 (1928) · JFM 54.0106.18
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