h-vector

In algebraic combinatorics, the h-vector of a simplicial polytope is a fundamental invariant of the polytope which encodes the number of faces of different dimensions and allows one to express the Dehn–Sommerville equations in a particularly simple form. A characterization of the set of h-vectors of simplicial polytopes was conjectured by Peter McMullen^[1] and proved by Lou Billera and Carl W. Lee^[2][3] and Richard Stanley^[4] (g-theorem). The definition of h-vector applies to arbitrary abstract simplicial complexes. The g-conjecture states that for simplicial spheres, all possible h-vectors occur already among the h-vectors of the boundaries of convex simplicial polytopes.

Stanley introduced a generalization of the h-vector, the toric h-vector, which is defined for an arbitrary ranked poset, and proved that for the class of Eulerian posets, the Dehn–Sommerville equations continue to hold. A different, more combinatorial, generalization of the h-vector that has been extensively studied is the flag h-vector of a ranked poset. For Eulerian posets, it can be more concisely expressed by means of a noncommutative polynomial in two variables called the cd-index.

Definition

Let Δ be an abstract simplicial complex of dimension d − 1 with f_i i-dimensional faces and f₋₁ = 1. These numbers are arranged into the f-vector of Δ,

$f(\Delta)=(f_{-1},f_0,\ldots,f_{d-1}).$

An important special case occurs when Δ is the boundary of a d-dimensional convex polytope.

For k = 0, 1, …, d, let

$h_k = \sum_{i=0}^k (-1)^{k-i}\binom{d-i}{k-i}f_{i-1}.$

The tuple

$h(\Delta)=(h_0,h_1,\ldots,h_d)$

is called the h-vector of Δ. The f-vector and the h-vector uniquely determine each other through the linear relation

$\sum_{i=0}^{d}f_{i-1}(t-1)^{d-i}= \sum_{k=0}^{d}h_{k}t^{d-k}.$

Let R = k[Δ] be the Stanley–Reisner ring of Δ. Then its Hilbert–Poincaré series can be expressed as

$P_{R}(t)=\sum_{i=0}^{d}\frac{f_{i-1}t^i}{(1-t)^{i}}= \frac{h_0+h_1t+\cdots+h_d t^d}{(1-t)^d}.$

This motivates the definition of the h-vector of a finitely generated positively graded algebra of Krull dimension d as the numerator of its Hilbert–Poincaré series written with the denominator (1 − t)^d.

The h-vector is closely related to the h^*-vector for a convex lattice polytope, see Ehrhart polynomial.

Toric h-vector

To an arbitrary graded poset P, Stanley associated a pair of polynomials f(P,x) and g(P,x). Their definition is recursive in terms of the polynomials associated to intervals [0,y] for all y ∈ P, y ≠ 1, viewed as ranked posets of lower rank (0 and 1 denote the minimal and the maximal elements of P). The coefficients of f(P,x) form the toric h-vector of P. When P is an Eulerian poset of rank d + 1 such that P − 1 is simplicial, the toric h-vector coincides with the ordinary h-vector constructed using the numbers f_i of elements of P − 1 of given rank i + 1. In this case the toric h-vector of P satisfies the Dehn–Sommerville equations

$h_k = h_{d-k}.$

The reason for the adjective "toric" is a connection of the toric h-vector with the intersection cohomology of a certain projective toric variety X whenever P is the boundary complex of rational convex polytope. Namely, the components are the dimensions of the even intersection cohomology groups of X:

$h_k=\operatorname{dim}_{\mathbb{Q}}\operatorname{IH}^{2k}(X,\mathbb{Q})$

(the odd intersection cohomology groups of X are all zero). The Dehn–Sommerville equations are a manifestation of the Poincaré duality in the intersection cohomology of X.

Flag h-vector and cd-index

A different generalization of the notions of f-vector and h-vector of a convex polytope has been extensively studied. Let P be a finite graded poset of rank n − 1, so that each maximal chain in P has length n. For any S, a subset of {1,…,n}, let α_P(S) denote the number of chains in P whose ranks constitute the set S. More formally, let

$|\cdot|: P\to\{0,1,\ldots,n\}$

be the rank function of P and let P_S be the S-rank selected subposet, which consists of the elements from P whose rank is in S:

$P_S=\{x\in P: |x|\in S\}.$

Then α_P(S) is the number of the maximal chains in P(S) and the function

$S\mapsto\alpha_P(S)$

is called the flag f-vector of P. The function

$S\mapsto\beta_P(S), \quad \beta_P(S)=\sum_{T\subseteq S}(-1)^{|S|-|T|}\alpha_P(S)$

is called the flag h-vector of P. By the inclusion–exclusion principle,

$\alpha_P(S)=\sum_{T\subseteq S}\beta_P(T).$

The flag f- and h-vectors of P refine the ordinary f- and h-vectors of its order complex Δ(P):

$f_{i-1}(\Delta(P))=\sum_{|S|=i}\alpha_P(S), \quad h_{i}(\Delta(P))=\sum_{|S|=i}\beta_P(S).$

The flag h-vector of P can be displayed via a polynomial in noncommutative variables a and b. For any subset S of {1,…,n}, define the corresponding monomial in a and b,

$u_S = u_1 \cdots u_n, \quad u_i=a \text{ for } i\notin S, u_i=b \text{ for } i\in S.$

Then the noncommutative generating function for the flag h-vector of P is defined by

$\Psi_P(a,b)=\sum_{S}\beta_P(S)u_{S}.$

From the relation between α_P(S) and β_P(S), the noncommutative generating function for the flag f-vector of P is

$\Psi_P(a,a+b)=\sum_{S}\alpha_P(S)u_{S}.$

Margaret Bayer and Lou Billera determined the most general linear relations that hold between the components of the flag h-vector of an Eulerian poset P. Fine noted an elegant way to state these relations: there exists a noncommutative polynomial Φ_P(c,d), called the cd-index of P, such that

$\Psi_P(a,b)=\Phi_P(a+b, ab+ba).$

Stanley proved that all coefficients of the cd-index of the boundary complex of a convex polytope are non-negative. He conjectured that this positivity phenomenon persists for a more general class of Eulerian posets that Stanley calls Gorenstein* complexes and which includes simplicial spheres and complete fans. This conjecture was proved by Kalle Karu.^[5] The combinatorial meaning of these non-negative coefficients (an answer to the question "what do they count?") remains unclear.

References

1. McMullen, Peter (1971), "The numbers of faces of simplicial polytopes", Israel Journal of Mathematics 9 (4): 559–570, doi:10.1007/BF02771471, MR 0278183.
2. Billera, Louis; Lee, Carl (1980), "Sufficiency of McMullen's conditions for f-vectors of simplicial polytopes", Bulletin of the American Mathematical Society 2 (1): 181–185, doi:10.1090/s0273-0979-1980-14712-6, MR 551759.
3. Billera, Louis; Lee, Carl (1981), "A proof of the sufficiency of McMullen's conditions for f-vectors of simplicial convex polytopes", Journal of Combinatorial Theory, Series A 31 (3): 237–255, doi:10.1016/0097-3165(81)90058-3.
4. Stanley, Richard (1980), "The number of faces of a simplicial convex polytope", Advances in Mathematics 35 (3): 236–238, doi:10.1016/0001-8708(80)90050-X, MR 0563925.
5. Karu, Kalle (2006), "The cd-index of fans and posets", Compositio Mathematica 142: 701–718, doi:10.1112/S0010437X06001928, MR 2231198.

Further reading

• Stanley, Richard (1996), Combinatorics and commutative algebra, Progress in Mathematics 41 (2nd ed.), Boston, MA: Birkhäuser Boston, Inc., ISBN 0-8176-3836-9.
• Stanley, Richard (1997), Enumerative Combinatorics 1, Cambridge University Press, ISBN 0-521-55309-1.