H vector - Alchetron, The Free Social Encyclopedia

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 states that for simplicial spheres, all possible h-vectors occur already among the h-vectors of the boundaries of convex simplicial polytopes.

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 ( Δ ) = ( f − 1 , f 0 , … , 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 = ∑ i = 0 k ( − 1 ) k − i ( d − i k − i ) f i − 1 .

The tuple

h ( Δ ) = ( h 0 , h 1 , … , h d )

is called the h-vector of Δ. The f-vector and the h-vector uniquely determine each other through the linear relation

∑ i = 0 d f i − 1 ( t − 1 ) d − i = ∑ k = 0 d h k t d − k .

Let R = k[Δ] be the Stanley–Reisner ring of Δ. Then its Hilbert–Poincaré series can be expressed as

P R ( t ) = ∑ i = 0 d f i − 1 t i ( 1 − t ) i = h 0 + h 1 t + ⋯ + 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.

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 Q ⁡ IH 2 k ⁡ ( X , 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.

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 − 1, so that each maximal chain in P has length n. For any S, a subset of {1,…,n}, let α_P(S) denote the number of chains in P whose ranks constitute the set S. More formally, let

| ⋅ | : P → { 0 , 1 , … , 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 ∈ P : | x | ∈ S } .

Then α_P(S) is the number of the maximal chains in P(S) and the function

S ↦ α P ( S )

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

S ↦ β P ( S ) , β P ( S ) = ∑ T ⊆ S ( − 1 ) | S | − | T | α P ( S )

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

α P ( S ) = ∑ T ⊆ S β P ( T ) .

The flag f- and h-vectors of P refine the ordinary f- and h-vectors of its order complex Δ(P):

f i − 1 ( Δ ( P ) ) = ∑ | S | = i α P ( S ) , h i ( Δ ( P ) ) = ∑ | S | = i β 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 ⋯ u n , u i = a for i ∉ S , u i = b for i ∈ S .

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

Ψ P ( a , b ) = ∑ S β 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

Ψ P ( a , a + b ) = ∑ S α P ( S ) u S .

Margaret Bayer and Lou 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

Ψ P ( a , b ) = Φ P ( a + b , a b + b a ) .

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.

References

H-vector Wikipedia

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Contents

Definition

Toric h-vector

Flag h-vector and cd-index

References