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 McMullen¹ and proved by Lou Billera and Carl W. Lee²³ and Richard Stanley⁴ (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.⁵⁶

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.

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,

${\displaystyle f(\Delta )=(f_{-1},f_{0},\ldots ,f_{d-1}).}$

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

For k = 0, 1, …, d, let

${\displaystyle h_{k}=\sum _{i=0}^{k}(-1)^{k-i}{\binom {d-i}{k-i}}f_{i-1}.}$

The tuple

${\displaystyle h(\Delta )=(h_{0},h_{1},\ldots ,h_{d})}$

is called the h-vector of Δ. In particular, ${\displaystyle h_{0}=1}$, ${\displaystyle h_{1}=f_{0}-d}$, and ${\displaystyle h_{d}=(-1)^{d}(1-\chi (\Delta ))}$, where ${\displaystyle \chi (\Delta )}$ is the Euler characteristic of ${\displaystyle \Delta }$. The f-vector and the h-vector uniquely determine each other through the linear relation

${\displaystyle \sum _{i=0}^{d}f_{i-1}(t-1)^{d-i}=\sum _{k=0}^{d}h_{k}t^{d-k},}$

from which it follows that, for ${\displaystyle i=0,\dotsc ,d}$,

${\displaystyle f_{i-1}=\sum _{k=0}^{i}{\binom {d-k}{i-k}}h_{k}.}$

In particular, ${\displaystyle f_{d-1}=h_{0}+h_{1}+\dotsb +h_{d}}$. Let R = k[Δ] be the Stanley–Reisner ring of Δ. Then its Hilbert–Poincaré series can be expressed as

${\displaystyle P_{R}(t)=\sum _{i=0}^{d}{\frac {f_{i-1}t^{i}}{(1-t)^{i}}}={\frac {h_{0}+h_{1}t+\cdots +h_{d}t^{d}}{(1-t)^{d}}}.}$

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.

The ${\displaystyle \textstyle h}$-vector ${\displaystyle (h_{0},h_{1},\dotsc ,h_{d})}$ can be computed from the ${\displaystyle \textstyle f}$-vector ${\displaystyle (f_{-1},f_{0},\dotsc ,f_{d-1})}$ by using the recurrence relation

${\displaystyle h_{0}^{i}=1,\qquad -1\leq i\leq d}$

${\displaystyle h_{i+1}^{i}=f_{i},\qquad -1\leq i\leq d-1}$

${\displaystyle h_{k}^{i}=h_{k}^{i-1}-h_{k-1}^{i-1},\qquad 1\leq k\leq i\leq d}$.

and finally setting ${\displaystyle \textstyle h_{k}=h_{k}^{d}}$ for ${\displaystyle \textstyle 0\leq k\leq d}$. For small examples, one can use this method to compute ${\displaystyle \textstyle h}$-vectors quickly by hand by recursively filling the entries of an array similar to Pascal's triangle. For example, consider the boundary complex ${\displaystyle \textstyle \Delta }$ of an octahedron. The ${\displaystyle \textstyle f}$-vector of ${\displaystyle \textstyle \Delta }$ is ${\displaystyle \textstyle (1,6,12,8)}$. To compute the ${\displaystyle \textstyle h}$-vector of ${\displaystyle \Delta }$, construct a triangular array by first writing ${\displaystyle d+2}$ ${\displaystyle \textstyle 1}$s down the left edge and the ${\displaystyle \textstyle f}$-vector down the right edge.

${\displaystyle {\begin{matrix}&&&&1&&&\\&&&1&&6&&\\&&1&&&&12&\\&1&&&&&&8\\1&&&&&&&&0\end{matrix}}}$

(We set ${\displaystyle f_{d}=0}$ 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:

${\displaystyle {\begin{matrix}&&&&1&&&\\&&&1&&6&&\\&&1&&5&&12&\\&1&&4&&7&&8\\1&&3&&3&&1&&0\end{matrix}}}$

The entries of the bottom row (apart from the final ${\displaystyle 0}$) are the entries of the ${\displaystyle \textstyle h}$-vector. Hence, the ${\displaystyle \textstyle h}$-vector of ${\displaystyle \textstyle \Delta }$ is ${\displaystyle \textstyle (1,3,3,1)}$.

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

${\displaystyle h_{k}=h_{d-k}.}$

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:

${\displaystyle h_{k}=\dim _{\mathbb {Q} }\operatorname {IH} ^{2k}(X,\mathbb {Q} )}$

(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. Let ${\displaystyle P}$ be a finite graded poset of rank n, so that each maximal chain in ${\displaystyle P}$ has length n. For any ${\displaystyle S}$, a subset of ${\displaystyle \left\{0,\ldots ,n\right\}}$, let ${\displaystyle \alpha _{P}(S)}$ denote the number of chains in ${\displaystyle P}$ whose ranks constitute the set ${\displaystyle S}$. More formally, let

${\displaystyle rk:P\to \{0,1,\ldots ,n\}}$

be the rank function of ${\displaystyle P}$ and let ${\displaystyle P_{S}}$ be the ${\displaystyle S}$-rank selected subposet, which consists of the elements from ${\displaystyle P}$ whose rank is in ${\displaystyle S}$:

${\displaystyle P_{S}=\{x\in P:rk(x)\in S\}.}$

Then ${\displaystyle \alpha _{P}(S)}$ is the number of the maximal chains in ${\displaystyle P_{S}}$ and the function

${\displaystyle S\mapsto \alpha _{P}(S)}$

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

${\displaystyle S\mapsto \beta _{P}(S),\quad \beta _{P}(S)=\sum _{T\subseteq S}(-1)^{|S|-|T|}\alpha _{P}(S)}$

is called the flag h-vector of ${\displaystyle P}$. By the inclusion–exclusion principle,

${\displaystyle \alpha _{P}(S)=\sum _{T\subseteq S}\beta _{P}(T).}$

The flag f- and h-vectors of ${\displaystyle P}$ refine the ordinary f- and h-vectors of its order complex ${\displaystyle \Delta (P)}$:⁸

${\displaystyle f_{i-1}(\Delta (P))=\sum _{|S|=i}\alpha _{P}(S),\quad h_{i}(\Delta (P))=\sum _{|S|=i}\beta _{P}(S).}$

The flag h-vector of ${\displaystyle P}$ can be displayed via a polynomial in noncommutative variables a and b. For any subset ${\displaystyle S}$ of {1,…,n}, define the corresponding monomial in a and b,

${\displaystyle u_{S}=u_{1}\cdots u_{n},\quad u_{i}=a{\text{ for }}i\notin S,u_{i}=b{\text{ for }}i\in S.}$

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

${\displaystyle \Psi _{P}(a,b)=\sum _{S}\beta _{P}(S)u_{S}.}$

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

${\displaystyle \Psi _{P}(a,a+b)=\sum _{S}\alpha _{P}(S)u_{S}.}$

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 Φ_P(c,d), called the cd-index of P, such that

${\displaystyle \Psi _{P}(a,b)=\Phi _{P}(a+b,ab+ba).}$

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.¹⁰ The combinatorial meaning of these non-negative coefficients (an answer to the question "what do they count?") remains unclear.