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 and proved by Lou Billera and Carl W. Lee and Richard Stanley. 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 f_i 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 of Δ. In particular,,, and, where is the Euler characteristic of. The f''-vector and the h-vector uniquely determine each other through the linear relation
from which it follows that, for,
In particular,. Let R = k 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 ^d.
The h-vector is closely related to the h^*-vector for a convex lattice polytope, see Ehrhart polynomial.

Recurrence relation

The -vector can be computed from the -vector by using the recurrence relation
and finally setting for. 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. For example, consider the boundary complex of an octahedron. The -vector of is. To compute the -vector of, construct a triangular array by first writing s down the left edge and the -vector down the right edge.
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 are the entries of the -vector. Hence, the -vector of is.

Toric ''h''-vector

To an arbitrary graded poset P, Stanley associated a pair of polynomials f and g. Their definition is recursive in terms of the polynomials associated to intervals for all y ∈ P, y ≠ 1, viewed as ranked posets of lower rank. The coefficients of f 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
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 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-vector of a convex polytope has been extensively studied. Let be a finite graded poset of rank n, so that each maximal chain in has length n. For any, a subset of, let denote the number of chains in whose ranks constitute the set. More formally, let
be the rank function of and let be the -rank selected subposet, which consists of the elements from whose rank is in :
Then is the number of the maximal chains in and the function
is called the flag f-vector of P. The function
is called the flag h-vector of. By the inclusion–exclusion principle,
The flag f- and h-vectors of refine the ordinary f- and h-vectors of its order complex :
The flag h-vector of can be displayed via a polynomial in noncommutative variables a and b. For any subset of, 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 α_P and β_P, 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 Φ_P, 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 remains unclear.