STANLEY’S SIMPLICIAL POSET CONJECTURE, AFTER M. MASUDA

EZRA MILLER AND VIC REINER

Abstract. M. Masuda recently provided the missing piece proving a conjecture of R.P. Stanley on the characterization of f-vectors for Gorenstein* simplicial posets. We propose a slight simplification of Masuda’s proof.

Our main result, Theorem 2, was first proved by Masuda [Mas03], completing the missing step in a conjecture of Stanley characterizing the f-vectors of Gorenstein* simplicial posets. This note gives a simplified proof of it, using elementary methods. We begin with some background on simplicial posets; see Stanley [Sta91] for more detail and explanations for assertions not justified here.

A simplicial poset P is a finite poset with a minimal element ˆ0 such that every interval [ˆ0, p] for p ∈ P is a boolean algebra. We shall work instead with the associated regular cell complex Γ = Γ(P), whose face poset is P. The (closed) faces of Γ are simplices that meet pairwise in subcomplexes of their boundaries [Sta91]. For simplicity, we identify each face G of Γ (denoted G ∈ Γ in what follows) with the corresponding element of P.

Let S = 𝕜[x_{G} : G ∈ Γ] be a polynomial ring over a field 𝕜 in indeterminates indexed by the faces of Γ. The face ring of Γ is the quotient A_{Γ} = S/I_{Γ}, where I_{Γ} = ⟨x_{G}x_{G′} − x_{G∧G′} ∑_{F} x_{F} ⟩. Here the summation runs over the minimal faces F among those containing both G and G′, and the meet G ∧ G′ is the largest face in Γ that is contained in both G and G′; its uniqueness (when the sum is nonzero) follows from the fact that G and G′ lie in the boolean algebra [ˆ0, F] for any common upper bound F in the sum.

Write f_{i} = f_{i}(Γ) for the number of faces of dimension i in Γ, and set f_{−1} = 1. Letting d − 1 be the dimension of Γ, one has an equivalent encoding of the f-vector (f_{−1}, f_{0}, f_{1}, …, f_{d−1}) via the h-vector (h_{0}, h_{1}, …, h_{d}), whose entries are uniquely defined by the equation

∑_{i=0}^{d} f_{i−1}(t − 1)^{d−i} = ∑_{i=0}^{d} h_{i}t^{d−i}.

Stanley [Sta91] completely characterized the possible f-vectors (or h-vectors) when Γ is Cohen-Macaulay over 𝕜, and almost characterized the possible f-vectors when Γ satisfies the stronger condition of being Gorenstein* over 𝕜, that is, when Γ triangulates a 𝕜-homology sphere. When Γ is Cohen-Macaulay, (h_{0}, h_{1}, …, h_{d}) can be interpreted as the Hilbert function of the quotient ring A_{Γ}/Θ, where Θ is the ideal generated by any linear system of parameters θ_{1}, …, θ_{d}. Consequently h_{i} ≥ 0 for all i = 0, …, d, and this nonnegativity is sufficient to characterize these h-vectors [Sta91, Theorem 3.10]. The stronger Gorenstein* property further implies that A_{Γ}/Θ will be a Poincaré duality algebra, and hence h_{i} = h_{d−i} for all i. This almost characterizes such h-vectors, as shown by the following theorem of Masuda, conjectured by Stanley.

Theorem 1. [Mas03, Corollary 1.2], [Sta91, Remark 5] Let (h_{0}, …, h_{d}) ∈ ℕ^{d+1} satisfy h_{i} = 1 and h_{i} = h_{d−i} for all i. Then there is a Gorenstein* simplicial poset P of rank d with h_{i} = h_{i}(Γ(P)) if and only if either

• h_{i} > 0 for all i = 0, …, d, or else

• ∑_{i=0}^{d} h_{i} is even.

Stanley proved half of this theorem, by showing that the above conditions on (h_{0}, …, h_{d}) are sufficient to explicitly construct such a Gorenstein* simplicial poset. For the other half, since f_{d−1} = ∑_{i=0}^{d} h_{i}, it only remains to show that the condition h_{i} = 0 for some i = 1, …, d−1 forces Γ to have an even number of (d−1)-dimensional faces (called facets). In fact, Masuda shows (see discussion following [Mas03, Eq. (5.1)]) that the assumption h_{i} = 0 implies the following stronger property.

Theorem 2. [Mas03] If Γ is Gorenstein* and h_{i}(Γ) = 0 for some i strictly between 0 and d, then for every subset V = {v_{1}, …, v_{d}} of vertices, the number of facets in Γ having vertex set V is even.

Proof. Since the quotient A_{Γ}/Θ is a Poincaré duality algebra, one has h_{d} = 1, that is, its degree d piece (its socle) is a 1-dimensional vector space over 𝕜. Using the relations in I_{Γ}, one can see that the product x_{v_{1}} ⋯ x_{v_{d}} in A_{Γ} is congruent to the sum ∑_{F} x_{F} as F ranges over all facets having V as their vertex set. If h_{i} = 0, then as observed in [Mas03], the image of x_{v_{1}} ⋯ x_{v_{d}} modulo Θ will be zero, because it has factors (such as x_{v_{1}} ⋯ x_{v_{i}}) lying in the vanishing i^{th}-graded component (A_{Γ}/Θ)_{i}. Hence the image of ∑_{F} x_{F} modulo Θ must be zero in (A_{Γ}/Θ)_{i}. Since this sum takes place in the socle of A_{Γ}/Θ, it is therefore enough to prove the following general claims:

(i) the images modulo Θ of the variables x_{F} for facets F containing the vertices v_{1}, …, v_{d} are all the same up to ± sign in the 1-dimensional socle, and

(ii) these images x̄_{F} are all nonzero. Indeed, since the sum ∑_{F} x_{F} modulo Θ is zero in the socle, (i) and (ii) would imply that there must be an even number of terms in the sum.

Claims (i) and (ii) follow from Proposition 5 and Corollary 7, below, since Γ being a 𝕜-homology sphere implies that it is also a pseudomanifold. □

The rest of this note proves results implying Claims (i) and (ii). We do not assume that Γ is Gorenstein* anywhere in what follows, unless explicitly stated otherwise.

For a monomial m = x_{G_{1}} ⋯ x_{G_{r}} in A_{Γ}, define m to be standard if G_{1} ⊆ ⋯ ⊆ G_{r} is a (weak) chain in Γ. The following is a slight strengthening of Stanley’s observation [Sta91, Lemma 3.9] that A_{Γ} is integral over its subalgebra generated by (A_{Γ})_{1}.

Lemma 3. The variables x_{G} for G ∈ Γ generate A_{Γ} as a module over the subalgebra of A_{Γ} generated by (A_{Γ})_{1}. In fact, A_{P} is spanned 𝕜-linearly by monomials mx_{G} in which m is a monomial in the variables x_{v} for the vertices v of G.

Proof. Every element of A_{Γ} is a sum of standard monomials by [Sta91, Lemma 3.4]. Let x_{G_{1}} ⋯ x_{G_{r}} be a standard monomial, so G_{1} ⊆ ⋯ ⊆ G_{r}. For each face G ∈ Γ, denote by x^{G} the product of all variables x_{v} for vertices v of G. Now, for every index i < r, use the defining relations of A_{Γ} to replace x_{G_{i}} with x^{G_{i}} − ∑ x_{G′}, the sum being over all minimal faces G′ ≠ G_{i} containing the vertices of G_{i}. Observe that x_{G′}x_{G_{r}} = 0 in A_{Γ} for the faces G′ in the sum, because no face contains both G_{r} and a face other than G_{i} with the same vertices as G_{i}. Hence x_{G_{1}} ⋯ x_{G_{r}} is equal to x_{G_{r}} times a monomial in the variables x_{v} for vertices v of G. □

Abusing notation slightly, let Θ denote a linear system of parameters θ_{1}, …, θ_{d} for A_{Γ}, and 𝕜[Θ] the polynomial subalgebra of A_{Γ} that they generate. After choosing an ordering on the vertices of Γ, one can express Θ as a d × n matrix whose rows are θ_{1}, …, θ_{d}. As observed by Masuda [Mas03, Lemma 3.1], given any facet F of Γ, one can compose the finite extension 𝕜[Θ] ↪ A_{Γ} with the surjection A_{Γ} ↠ A_{[ˆ0,F]} that sends all variables x_{G} for G ⊄ F to zero. Because the composite 𝕜[Θ] → A_{[ˆ0,F]} must also be finite, and since A_{[ˆ0,F]} is a polynomial ring on the variables {x_{v}}_{v_{∈}F}, the d × d submatrix Θ_{F} of Θ with columns indexed by vertices in F has nonzero determinant det(Θ_{F}). For any element y ∈ A_{Γ}, denote by ȳ the image of y in the quotient ring A_{Γ}/Θ.

Lemma 4. (cf. [Ful93, §5.2 Lemma, p. 107]) Suppose that Γ is pure, meaning that its facets all have dimension d − 1. Let m be a monomial in the variables x_{v} for vertices v in a facet F. Then, for any face G of F, the image m̄x̄_{G} of mx_{G} in A_{Γ}/Θ equals a sum of terms m̄′x̄_{G} in which each monomial m′ is a product of variables x_{v} for distinct vertices v outside of G.

Proof. Invertibility of Θ_{F} implies that for any vertex v of F, the 𝕜-span of θ_{1}, …, θ_{d} contains a linear form θ′ = x_{v} + ∑_{w∉F} c_{w}x_{w} for some constants c_{w} ∈ 𝕜.

We first find a sum as in the lemma in which each m′ is only squarefree (but may involve vertices of G), by induction on the sum of all exponents ≥ 2 on variables x_{v} in m. Suppose x_{v}^{2} divides m, and write m = x_{v}ℓ. Use the linear form θ′ to write

m̄x̄_{G} = − ∑_{w∉F} c_{w}x̄_{w}ℓ̄x̄_{G}.

If x̄_{w}ℓ̄x̄_{G} ≠ 0, then some facet F′ containing G also contains all vertices appearing in x_{w}ℓ. By induction, x̄_{w}ℓ̄x̄_{G} can be rewritten as desired.

Now assume that m is squarefree, and use a similar argument, this time by induction on the number of variables x_{v} dividing m for vertices v ∈ G. The fact that each w in θ′ is not in F ensures that we re-create the squarefree hypothesis at each stage. □

Proposition 5. If Γ is pure, the images in A_{Γ}/Θ of the variables x_{G} for faces G ∈ Γ span 𝕜-linearly. In particular, if Γ is Gorenstein*, then x̄_{F} ≠ 0 in A_{Γ}/Θ for some facet F.

Proof. By Lemmas 3 and 4, every element of A_{Γ} can be expressed mod Θ as a sum of monomials of the form m′x_{G′}, where m′ is a product of variables x_{v} for distinct vertices v ∉ G′. But in A_{Γ}, such a monomial m′x_{G′} equals the sum of the variables x_{G} as G runs over all faces minimal with respect to the property that they contain both G′ and all vertices v for which x_{v} divides m′. □

Say that two facets F, F′ in a pure (d − 1)-dimensional complex share a thin ridge if their intersection is a (d − 2)-face contained in no other (d − 1)-faces.

Proposition 6. When Γ is pure, any two facets F, F′ sharing a thin ridge will have x̄_{F} = ±\frac{det(Θ_{F′})}{det(Θ_{F})}x̄_{F′} in A_{Γ}/Θ.

Proof. For convenience of notation, denote \frac{det(Θ_{F′})}{det(Θ_{F})} by δ. Let G = F ∧ F′, and let v_{F} , v_{F′} be their corresponding vertices not in G. There are two possibilities for these vertices: either v_{F} = v_{F′} or else v_{F} ≠ v_{F′}.

When v_{F} = v_{F′} = v, we get x_{v}x_{G} = x_{F} + x_{F′}. The argument in Lemma 4 shows that, modulo Θ, we can write x̄_{v} as a linear combination of degree 1 elements x̄_{v} for vertices v not in F. Since G lies only in the facets F and F′, this implies that x̄_{F} + x̄_{F′} = x̄_{v}x̄_{G} = 0. Hence in this case, x̄_{F} = −x̄_{F′} (which equals −δx̄_{F′}, since Θ_{F′} = Θ_{F}).

When v_{F} ≠ v_{F′}, we get x_{v_{F}} x_{G} = x_{F}, and x_{v_{F′}}x_{G} = x_{F′}. After multiplying the matrix Θ on the left by Θ_{F}^{−1}, one of the rows gives (by Cramer’s rule) a linear form x_{v_{F}} ∓ δx_{v_{F′}} + ∑ c_{v}x_{v} in the ideal Θ, where the sum is over all vertices v ≠ v_{F′} that do not lie in F, and the c_{v} are constants in 𝕜. Hence

x̄_{F} = x̄_{v_{F}} x̄_{G} = (±δx_{v_{F′}} − ∑ c_{v}x_{v})x̄_{G} = ±δx̄_{v_{F′}} x̄_{G} = ±δx̄_{F′},

where the third equality holds because x_{v}x_{G} = 0 whenever v does not lie in F ∪F′. □

Corollary 7. Assume Γ is pure and F, F′ are facets with the same set of vertices that are connected by a sequence F = F_{1}, …, F_{r} = F′ in which F_{j} and F_{j−1} share a thin ridge for 2 ≤ j ≤ r. Then x̄_{F} = ±x̄_{F′}.

Proof. Use Proposition 6: the product of ratios of determinants telescopes to ±1. □

Example 8. Let Γ be obtained by slitting a hollow tetrahedron along a single edge, and attaching a “pita pocket” of two triangles sewn together along two of their common edges. Suppose that the two pita triangles have vertices 123, while the slit tetrahedron has vertices 2345. Take a path from from the top pita triangle to the bottom pita triangle by traversing the facets with vertices 234, then 345, then 235, and finally back to the bottom pita triangle. Writing [ijk] = det(Θ_{F}) when F has vertex set ijk (these are Plücker coordinates), the sequence of ratios of determinants is [234]/[123], then [345]/[234], then −[235]/[345], and finally [123]/[235]. Note that the product of all these is −1, which is also the sign obtained by flipping from the top pita triangle to the bottom one along one of the two codimension 1 faces they share.

References

[Ful93] W. Fulton, Introduction to toric varieties. Annals of Mathematics Studies 131. Princeton University Press, Princeton, NJ, 1993.

[Mas03] Mikiya Masuda, h-vectors of Gorenstein* simplicial posets, 2003. arXiv:math.CO/0305203 [Sta91] Richard P. Stanley, f-vectors and h-vectors of simplicial posets, J. Pure Appl. Algebra 71

(1991), no. 2-3, 319–331.