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 ("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 "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. 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.
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
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"-vector of a convex polytope has been extensively studied. Let formula_8 be a finite graded poset of rank "n", so that each maximal chain in formula_8 has length "n". For any formula_10, a subset of formula_11, let formula_12 denote the number of chains in formula_8 whose ranks constitute the set formula_10. More formally, let
be the rank function of formula_8 and let formula_17 be the formula_10-rank selected subposet, which consists of the elements from formula_8 whose rank is in formula_10:
Then formula_12 is the number of the maximal chains in formula_17 and the function
is called the flag "f"-vector of "P". The function
is called the flag "h"-vector of formula_8. By the inclusion–exclusion principle,
The flag "f"- and "h"-vectors of formula_8 refine the ordinary "f"- and "h"-vectors of its order complex formula_29:
The flag "h"-vector of formula_8 can be displayed via a polynomial in noncommutative variables "a" and "b". For any subset formula_10 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 "α""P"("S") and "β""P"("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 Φ"P"("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.