f-vectors and h-vectors of simplicial posets. (English) Zbl 0727.06009

Stanley, Richard P.

J. Pure Appl. Algebra 71, No. 2-3, 319-331 (1991).

A simplicial poset is a finite poset P with \({\hat 0}\) such that every interval \([{\hat 0},x]\) is a boolean algebra. Simplicial posets are generalizations of simplicial complexes. Like degree sequences for graphs f-vectors are associated with simplicial posets. Thus, if P is a simplicial poset of rank d, then \(f(P)=(f_ 0,f_ 1,...,f_{d- 1})=| \{x\in P|\) \([{\hat 0},x]\cong B_{i+1}\}|\), where \(B_{i+1}\) is a boolean algebra of rank \(i+1\). The author shows that (Theorem 2.1) if \(f=(f_ 0,...,f_{d-1})\in {\mathbb{Z}}^ d\), then \(f=f(P)\) for a simplicial poset iff \(f_ i\geq \binom{d}{i+1}\) for \(1\leq i\leq d-1\). It seems that \(| \{P|\) \(f=f(P)\}| =S(f)\) (say, the Stanley-number of f), where P is a simplicial poset and where \(P=Q\) iff P and Q are isomorphic, is in fact an interesting number deserving further discussion. Since with \(f=(f_ 0,...,f_{d-1})\) one may associate a monomial \(M(f)=x_ 0^{f_ 0}...x_{d-1}^{f_{d-1}}\), there is an associated generating function \({\mathcal S}(x_ 0,x_ 1,...)=\sum_{f}S(f)M(f)\) (say, the Stanley-function), whose properties should be interesting and which might be analysable using the methods for which the author of this paper is well-known indeed. The h-vector is defined from the f-vector by \(\sum^{d}_{i=0}f_{i-1}(x-1)^{d- i}=\sum^{d}_{i=0}h_ ix^{d-i}.\) Then h-vectors have been characterized for Cohen-Macaulay simplicial complexes. In (Theorem 3.10) it is shown that if \(h=(h_ 0,...,h_ d)\in {\mathbb{Z}}^{d+1}\) then \(h=h(P)\) for a Cohen-Macaulay simplicial poset iff \(h_ 0=1\) and \(h_ i\geq 0\) for all i. For Gorenstein posets and the derived notion Gorenstein* poset sufficient conditions on the h-vector are also provided (Theorem 4.3), with further conditions conditioning necessity. The methods of proof are (co)homological-algebraic-combinatorial in nature in the clear style for which the author is well-known.

Reviewer: J.Neggers (Tuscaloosa)

MSC:

06A11	Algebraic aspects of posets
05E99	Algebraic combinatorics
52B20	Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)

Keywords:

Cohen-Macaulay ring; Gorenstein complex; simplicial poset; simplicial complexes; f-vectors; Stanley-number; Stanley-function; Cohen-Macaulay simplicial complexes; Cohen-Macaulay simplicial poset; Gorenstein posets; h-vector

References:

[1]	Atiyah, M.F.; MacDonald, I.G., Introduction to commutative algebra, (1969), Addison-Wesley Reading, MA · Zbl 0175.03601
[2]	Baclawski, K., Rings with lexicographic straightening law, Adv. in math., 39, 185-213, (1981) · Zbl 0466.13004
[3]	Björner, A., Posets, regular CW complexes and Bruhat order, European J. combin., 5, 7-16, (1984) · Zbl 0538.06001
[4]	Björner, A., Face numbers of complexes and polytopes, Proceedings intenational congress of mathematicians, 1408-1418, (1987), Berkeley, 1986
[5]	Björner, A.; Frankl, P.; Stanley, R., The number of faces of balanced Cohen-Macaulay complexes and a generalized Macaulay theorem, Combinatorica, 7, 23-34, (1987) · Zbl 0651.05010
[6]	Björner, A.; Garsia, A.; Stanley, R., An introduction to the theory of Cohen-Macaulay partially ordered sets, (), 583-615
[7]	DeConcini, C.; Eisenbud, D.; Procesi, C., Hodge algebras, Astérisque, 91, 1-87, (1982)
[8]	Eisenbud, D., Introduction to algebras with straightening laws, (), 243-268
[9]	Garsia, A.M.; Stanton, D., Group actions on Stanley-Reisner rings and invariants of permutation groups, Adv. in math., 51, 107-201, (1984) · Zbl 0561.06002
[10]	Hibi, T., Distributive lattices, semigroup rings and algebras with straightening laws, (), 93-109
[11]	Kaplansky, I., Commutative rings, (1970), Allyn and Bacon Boston, MA · Zbl 0203.34601
[12]	Matsumura, H., Commutative ring theory, (1986), Cambridge Univ. Press Cambridge
[13]	Stanley, R., Cohen-Macaulay complexes, (), 51-62
[14]	Stanley, R., Hilbert functions of graded algebras, Adv. in math., 28, 57-83, (1978) · Zbl 0384.13012
[15]	Stanley, R., Combinatorics and commutative algebra, () · Zbl 0838.13008
[16]	Stanley, R., Enumerative combinatorics, Vol. I, (1986), Wadsworth and Brooks/Cole Monterey, CA · Zbl 0608.05001