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 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.
[5][6] 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.
[citation needed] 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.
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.
The f-vector and the h-vector uniquely determine each other through the linear relation from which it follows that, for
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.
For small examples, one can use this method to compute
-vectors quickly by hand by recursively filling the entries of an array similar to Pascal's triangle.
, construct a triangular array by first writing
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: The entries of the bottom row (apart from the final
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.
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.
[7] A different generalization of the notions of f-vector and h-vector of a convex polytope has been extensively studied.
be a finite graded poset of rank n, so that each maximal chain in
-rank selected subposet, which consists of the elements from
refine the ordinary f- and h-vectors of its order complex
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.[9] 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.
[10] The combinatorial meaning of these non-negative coefficients (an answer to the question "what do they count?")