H-vector

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title: "H-vector" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["algebraic-combinatorics", "polyhedral-combinatorics"] topic_path: "science/mathematics" source: "https://en.wikipedia.org/wiki/H-vector" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

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{{citation | last = McMullen | first = Peter | author-link = Peter McMullen | mr = 0278183 | title = The numbers of faces of simplicial polytopes | journal = Israel Journal of Mathematics | volume = 9 | issue = 4 | pages = 559–570 | year = 1971 | doi = 10.1007/BF02771471| s2cid = 92984501 }}. and proved by Lou Billera and Carl W. Lee{{citation | last1 = Billera | first1 = Louis | author-link1 = Louis Billera | last2 = Lee | first2 = Carl | mr = 551759 | title = Sufficiency of McMullen's conditions for f-vectors of simplicial polytopes | journal = Bulletin of the American Mathematical Society | volume = 2 | issue = 1 | pages = 181–185 | year = 1980 | doi=10.1090/s0273-0979-1980-14712-6| doi-access = free | last1 = Billera | first1 = Louis | author-link1 = Louis Billera | last2 = Lee | first2 = Carl | title = A proof of the sufficiency of McMullen's conditions for f-vectors of simplicial convex polytopes | journal = Journal of Combinatorial Theory, Series A | volume = 31 | issue = 3 | pages = 237–255 | year = 1981 | doi = 10.1016/0097-3165(81)90058-3| doi-access = free | last = Stanley | first = Richard | author-link = Richard P. Stanley | mr = 0563925 | title = The number of faces of a simplicial convex polytope | journal = Advances in Mathematics | volume = 35 | issue = 3 | pages = 236–238 | year = 1980 | doi = 10.1016/0001-8708(80)90050-X| doi-access=free}}. (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,

: 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

: h_k = \sum_{i=0}^k (-1)^{k-i}\binom{d-i}{k-i}f_{i-1}.

The tuple

: h(\Delta)=(h_0,h_1,\ldots,h_d)

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

: \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 i = 0, \dotsc, d,

:f_{i-1} = \sum_{k=0}^i \binom{d-k}{i-k} h_{k}.

In particular, 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

: P_{R}(t)=\sum_{i=0}^{d}\frac{f_{i-1}t^i}{(1-t)^{i}}= \frac{h_0+h_1t+\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.

Recurrence relation

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

:h^{i}{0} = 1, \qquad -1 \le i \le d :h^{i}{i+1} = f_{i}, \qquad -1 \le i \le d-1 :h^{i}{k} = h^{i-1}{k} - h^{i-1}_{k-1}, \qquad 1 \le k \le i \le d.

and finally setting \textstyle h_{k} = h^{d}_{k} for \textstyle 0 \le k \le d. For small examples, one can use this method to compute \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 \textstyle \Delta of an octahedron. The \textstyle f-vector of \textstyle \Delta is \textstyle (1, 6, 12, 8). To compute the \textstyle h-vector of \Delta, construct a triangular array by first writing d+2 \textstyle 1s down the left edge and the \textstyle f-vector down the right edge.

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

(We set 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:

:\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 0) are the entries of the \textstyle h-vector. Hence, the \textstyle h-vector of \textstyle \Delta is \textstyle (1, 3, 3, 1).

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

: 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:

: 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.

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

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

be the rank function of P and let P_S be the S-rank selected subposet, which consists of the elements from P whose rank is in S:

: P_S={x\in P: rk(x)\in S}.

Then \alpha_P(S) is the number of the maximal chains in P_S and the function

: S \mapsto \alpha_P(S)

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

: 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 P. By the inclusion–exclusion principle,

: \alpha_P(S) = \sum_{T\subseteq S}\beta_P(T).

The flag f- and h-vectors of P refine the ordinary f- and h-vectors of its order complex \Delta(P):{{citation | last = Stanley | first = Richard | doi = 10.2307/1998915 | journal = Transactions of the American Mathematical Society | pages = 139157 | title = Balanced Cohen-Macaulay Complexes | volume = 249 | number = 1 | year = 1979| jstor = 1998915 | doi-access = free

: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 P can be displayed via a polynomial in noncommutative variables a and b. For any subset S of {1,…,n}, define the corresponding monomial in a and b,

: 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

: \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

: \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.

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

: \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.{{citation | last = Karu | first = Kalle | mr = 2231198 | doi = 10.1112/S0010437X06001928 | journal = Compositio Mathematica | pages = 701718 | title = The cd-index of fans and posets | volume = 142 | issue = 3 | year = 2006| doi-access = free

References

References

1. Kalai, Gil. (2018-12-25). "Amazing: Karim Adiprasito proved the g-conjecture for spheres!".
2. Adiprasito, Karim. (2018-12-26). "Combinatorial Lefschetz theorems beyond positivity".
3. Karu, Kalle. (2004-08-01). "Hard Lefschetz theorem for nonrational polytopes". Inventiones Mathematicae.
4. Bayer, Margaret M. and Billera, Louis J (1985), "Generalized Dehn-Sommerville relations for polytopes, spheres and Eulerian partially ordered sets", Inventiones Mathematicae '''79''': 143-158. doi:10.1007/BF01388660.

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