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 DehnâÂÂSommerville 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 DehnâÂÂSommerville 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.
Let àbe an abstract simplicial complex of dimension d â 1 with f<sub>i</sub> i-dimensional faces and f<sub>âÂÂ1</sub> = 1. These numbers are arranged into the f-vector of ÃÂ,
An important special case occurs when ÃÂ is the boundary of a d-dimensional convex polytope.
For k = 0, 1, â¦, d, let
The tuple
is called the h-vector of ÃÂ. 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[ÃÂ] be the StanleyâÂÂReisner ring of ÃÂ. Then its HilbertâÂÂPoincaré 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 HilbertâÂÂPoincaré series written with the denominator (1 â t)<sup>d</sup>.
The h-vector is closely related to the h<sup>*</sup>-vector for a convex lattice polytope, see Ehrhart polynomial.
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.
(We set 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:
The entries of the bottom row (apart from the final ) are the entries of the -vector. Hence, the -vector of is .
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 â P, y â 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 â 1 is simplicial, the toric h-vector coincides with the ordinary h-vector constructed using the numbers f<sub>i</sub> of elements of P â 1 of given rank i + 1. In this case the toric h-vector of P satisfies the DehnâÂÂSommerville 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 DehnâÂÂSommerville 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.
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 inclusionâÂÂexclusion 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 {1,â¦,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 ñ<sub>P</sub>(S) and ò<sub>P</sub>(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 æ<sub>P</sub>(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.