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 and proved by Lou Billera and Carl W. Lee and Richard Stanley (g-theorem). The definition of h-vector applies to arbitrary abstract simplicial complexes. The g-conjecture stated that for simplicial spheres, all possible h-vectors occur already among the h-vectors of the boundaries of convex simplicial polytopes. It was proven in December 2018 by Karim Adiprasito.

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 fi i-dimensional faces and f−1 = 1. These numbers are arranged into the f-vector of Δ,

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

For k = 0, 1, …, d, let

The tuple

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

,

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

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.

Recurrence relation

by using the recurrence relation

1. .

-vectors quickly by hand by recursively filling the entries of an array similar to Pascal's triangle of an octahedron-vector down the right edge.

just to make the array triangular.) Then, starting from the top, fill each remaining entry by subtracting its upper-left neighbor from its upper-right neighbor. In this way, we generate the following array:

.

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 fi of elements of P − 1 of given rank i + 1. In this case the toric h-vector of P satisfies the Dehn–Sommerville equations

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:

(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. Kalle Karu proved that the toric h-vector of a polytope is unimodal, regardless of whether the polytope is rational or not.

Flag h-vector and cd-index

A different generalization of the notions of f-vector and h be a finite graded poset of rank n, so that each maximal chain has length n. More formally, let

be the -rank selected subposet:

and the function

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

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

The flag f- and h refine the ordinary f- and h-vectors of its order complex:

The flag h can be displayed via a polynomial in noncommutative variables a and b of {1,…,n}, define the corresponding monomial in a and b,

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

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

Margaret Bayer and Louis 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 ΦP(c,d), called the cd-index of P, such that

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. The combinatorial meaning of these non-negative coefficients (an answer to the question "what do they count?") remains unclear.

References

Further reading

• .

• .