A generalized Macaulay theorem and generalized face rings

Eran Nevo

doi:10.1016/j.jcta.2005.12.002

A generalized Macaulay theorem and generalized face rings

Eran Nevo^*

^*Corresponding author for this work

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

We prove that the f-vector of members in a certain class of meet semi-lattices satisfies Macaulay inequalities 0 ≤ ∂^k (f_k) ≤ f_{k - 1} for all k ≥ 0. We construct a large family of meet semi-lattices belonging to this class, which includes all posets of multicomplexes, as well as meet semi-lattices with the "diamond property," discussed by Wegner [G. Wegner, Kruskal-Katona's theorem in generalized complexes, in: Finite and Infinite Sets, vol. 2, in: Colloq. Math. Soc. János Bolyai, vol. 37, North-Holland, Amsterdam, 1984, pp. 821-828], as special cases. Specializing the proof to the later family, one obtains the Kruskal-Katona inequalities and their proof as in [G. Wegner, Kruskal-Katona's theorem in generalized complexes, in: Finite and Infinite Sets, vol. 2, in: Colloq. Math. Soc. János Bolyai, vol. 37, North-Holland, Amsterdam, 1984, pp. 821-828]. For geometric meet semi-lattices we construct an analogue of the exterior face ring, generalizing the classic construction for simplicial complexes. For a more general class, which also includes multicomplexes, we construct an analogue of the Stanley-Reisner ring. These two constructions provide algebraic counterparts (and thus also algebraic proofs) of Kruskal-Katona's and Macaulay's inequalities for these classes, respectively.

Original language	English
Pages (from-to)	1321-1331
Number of pages	11
Journal	Journal of Combinatorial Theory. Series A
Volume	113
Issue number	7
DOIs	https://doi.org/10.1016/j.jcta.2005.12.002
State	Published - Oct 2006

Bibliographical note

Funding Information:
I thank my advisor Prof. Gil Kalai for many helpful discussions, and Prof. Anders Björner for his comments on earlier versions of this paper. Part of this work was done during the author’s stay at Institut Mittag-Leffler, supported by the ACE network. I thank the referees, especially one of them, for their suggestions, which improved the presentation in this paper.

Keywords

Face ring
Kruskal-Katona
Macaulay inequalities
Meet semi-lattice
Shadow

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10.1016/j.jcta.2005.12.002

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title = "A generalized Macaulay theorem and generalized face rings",

abstract = "We prove that the f-vector of members in a certain class of meet semi-lattices satisfies Macaulay inequalities 0 ≤ ∂k (fk) ≤ fk - 1 for all k ≥ 0. We construct a large family of meet semi-lattices belonging to this class, which includes all posets of multicomplexes, as well as meet semi-lattices with the {"}diamond property,{"} discussed by Wegner [G. Wegner, Kruskal-Katona's theorem in generalized complexes, in: Finite and Infinite Sets, vol. 2, in: Colloq. Math. Soc. J{\'a}nos Bolyai, vol. 37, North-Holland, Amsterdam, 1984, pp. 821-828], as special cases. Specializing the proof to the later family, one obtains the Kruskal-Katona inequalities and their proof as in [G. Wegner, Kruskal-Katona's theorem in generalized complexes, in: Finite and Infinite Sets, vol. 2, in: Colloq. Math. Soc. J{\'a}nos Bolyai, vol. 37, North-Holland, Amsterdam, 1984, pp. 821-828]. For geometric meet semi-lattices we construct an analogue of the exterior face ring, generalizing the classic construction for simplicial complexes. For a more general class, which also includes multicomplexes, we construct an analogue of the Stanley-Reisner ring. These two constructions provide algebraic counterparts (and thus also algebraic proofs) of Kruskal-Katona's and Macaulay's inequalities for these classes, respectively.",

keywords = "Face ring, Kruskal-Katona, Macaulay inequalities, Meet semi-lattice, Shadow",

author = "Eran Nevo",

note = "Funding Information: I thank my advisor Prof. Gil Kalai for many helpful discussions, and Prof. Anders Bj{\"o}rner for his comments on earlier versions of this paper. Part of this work was done during the author{\textquoteright}s stay at Institut Mittag-Leffler, supported by the ACE network. I thank the referees, especially one of them, for their suggestions, which improved the presentation in this paper.",

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N2 - We prove that the f-vector of members in a certain class of meet semi-lattices satisfies Macaulay inequalities 0 ≤ ∂k (fk) ≤ fk - 1 for all k ≥ 0. We construct a large family of meet semi-lattices belonging to this class, which includes all posets of multicomplexes, as well as meet semi-lattices with the "diamond property," discussed by Wegner [G. Wegner, Kruskal-Katona's theorem in generalized complexes, in: Finite and Infinite Sets, vol. 2, in: Colloq. Math. Soc. János Bolyai, vol. 37, North-Holland, Amsterdam, 1984, pp. 821-828], as special cases. Specializing the proof to the later family, one obtains the Kruskal-Katona inequalities and their proof as in [G. Wegner, Kruskal-Katona's theorem in generalized complexes, in: Finite and Infinite Sets, vol. 2, in: Colloq. Math. Soc. János Bolyai, vol. 37, North-Holland, Amsterdam, 1984, pp. 821-828]. For geometric meet semi-lattices we construct an analogue of the exterior face ring, generalizing the classic construction for simplicial complexes. For a more general class, which also includes multicomplexes, we construct an analogue of the Stanley-Reisner ring. These two constructions provide algebraic counterparts (and thus also algebraic proofs) of Kruskal-Katona's and Macaulay's inequalities for these classes, respectively.

AB - We prove that the f-vector of members in a certain class of meet semi-lattices satisfies Macaulay inequalities 0 ≤ ∂k (fk) ≤ fk - 1 for all k ≥ 0. We construct a large family of meet semi-lattices belonging to this class, which includes all posets of multicomplexes, as well as meet semi-lattices with the "diamond property," discussed by Wegner [G. Wegner, Kruskal-Katona's theorem in generalized complexes, in: Finite and Infinite Sets, vol. 2, in: Colloq. Math. Soc. János Bolyai, vol. 37, North-Holland, Amsterdam, 1984, pp. 821-828], as special cases. Specializing the proof to the later family, one obtains the Kruskal-Katona inequalities and their proof as in [G. Wegner, Kruskal-Katona's theorem in generalized complexes, in: Finite and Infinite Sets, vol. 2, in: Colloq. Math. Soc. János Bolyai, vol. 37, North-Holland, Amsterdam, 1984, pp. 821-828]. For geometric meet semi-lattices we construct an analogue of the exterior face ring, generalizing the classic construction for simplicial complexes. For a more general class, which also includes multicomplexes, we construct an analogue of the Stanley-Reisner ring. These two constructions provide algebraic counterparts (and thus also algebraic proofs) of Kruskal-Katona's and Macaulay's inequalities for these classes, respectively.

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JO - Journal of Combinatorial Theory. Series A

JF - Journal of Combinatorial Theory. Series A

IS - 7

ER -

A generalized Macaulay theorem and generalized face rings

Abstract

Bibliographical note

Keywords

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