Английская Википедия:H-vector

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Шаблон:Lowercase 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,

<math> f(\Delta)=(f_{-1},f_0,\ldots,f_{d-1}).</math>

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

For k = 0, 1, …, d, let

<math> h_k = \sum_{i=0}^k (-1)^{k-i}\binom{d-i}{k-i}f_{i-1}. </math>

The tuple

<math> h(\Delta)=(h_0,h_1,\ldots,h_d) </math>

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

<math> \sum_{i=0}^{d}f_{i-1}(t-1)^{d-i}= \sum_{k=0}^{d}h_{k}t^{d-k}, </math>

from which it follows that, for <math>i = 0, \dotsc, d</math>,

<math>f_{i-1} = \sum_{k=0}^i \binom{d-k}{i-k} h_{k}.</math>

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

<math> 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}. </math>

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

<math>h^{i}_{0} = 1, \qquad -1 \le i \le d</math>

<math>h^{i}_{i+1} = f_{i}, \qquad -1 \le i \le d-1</math>

<math>h^{i}_{k} = h^{i-1}_{k} - h^{i-1}_{k-1}, \qquad 1 \le k \le i \le d</math>.

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

<math>\begin{matrix} & & & & 1 & & & \\ & & & 1 & & 6 & & \\ & & 1 & & & & 12 & \\ & 1 & & & & & & 8 \\ 1 & & & & & & & & 0 \end{matrix}</math>

(We set <math>f_{d} = 0</math> 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:

<math>\begin{matrix} & & & & 1 & & & \\ & & & 1 & & 6 & & \\ & & 1 & & 5 & & 12 & \\ & 1 & & 4 & & 7 & & 8 \\ 1 & & 3 & & 3 & & 1 & & 0 \end{matrix}</math>

The entries of the bottom row (apart from the final <math>0</math>) are the entries of the <math>\textstyle h</math>-vector. Hence, the <math>\textstyle h</math>-vector of <math>\textstyle \Delta</math> is <math>\textstyle (1, 3, 3, 1)</math>.

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

<math> h_k = h_{d-k}. </math>

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:

<math> h_k = \dim_{\mathbb{Q}} \operatorname{IH}^{2k}(X,\mathbb{Q}) </math>

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

<math> rk: P\to\{0,1,\ldots,n\}</math>

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

<math> P_S=\{x\in P: rk(x)\in S\}.</math>

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

<math> S \mapsto \alpha_P(S) </math>

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

<math> S \mapsto \beta_P(S), \quad

\beta_P(S) = \sum_{T \subseteq S} (-1)^{|S|-|T|} \alpha_P(S) </math>

is called the flag h-vector of <math>P</math>. By the inclusion–exclusion principle,

<math> \alpha_P(S) = \sum_{T\subseteq S}\beta_P(T). </math>

The flag f- and h-vectors of <math>P</math> refine the ordinary f- and h-vectors of its order complex <math>\Delta(P)</math>:

<math>f_{i-1}(\Delta(P)) = \sum_{|S|=i} \alpha_P(S), \quad

h_{i}(\Delta(P)) = \sum_{|S|=i} \beta_P(S). </math>

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

<math> u_S = u_1 \cdots u_n, \quad

u_i=a \text{ for } i\notin S, u_i=b \text{ for } i\in S. </math>

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

<math>\Psi_P(a,b) = \sum_{S} \beta_P(S) u_{S}. </math>

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

<math> \Psi_P(a,a+b) = \sum_{S} \alpha_P(S) u_{S}. </math>

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

<math> \Psi_P(a,b) = \Phi_P(a+b, ab+ba). </math>

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.

Further reading

• Шаблон:Citation.
• Шаблон:Citation.