h-vector

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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[1] and proved by Lou Billera and Carl W. Lee[2][3] and Richard Stanley[4] (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[edit]

Let Δ be an abstract simplicial complex of dimension d − 1 with fi 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[edit]

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 yP, 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 fi 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.

Flag h-vector and cd-index[edit]

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 αP(S) denote the number of chains in P whose ranks constitute the set S. More formally, let

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

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

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

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

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

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,

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

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

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 ΦP(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.[5] The combinatorial meaning of these non-negative coefficients (an answer to the question "what do they count?") remains unclear.

References[edit]

  1. ^ McMullen, Peter (1971), "The numbers of faces of simplicial polytopes", Israel Journal of Mathematics, 9 (4): 559–570, doi:10.1007/BF02771471, MR 0278183 .
  2. ^ Billera, Louis; Lee, Carl (1980), "Sufficiency of McMullen's conditions for f-vectors of simplicial polytopes", Bulletin of the American Mathematical Society, 2 (1): 181–185, doi:10.1090/s0273-0979-1980-14712-6, MR 551759 .
  3. ^ Billera, Louis; Lee, Carl (1981), "A proof of the sufficiency of McMullen's conditions for f-vectors of simplicial convex polytopes", Journal of Combinatorial Theory, Series A, 31 (3): 237–255, doi:10.1016/0097-3165(81)90058-3 .
  4. ^ Stanley, Richard (1980), "The number of faces of a simplicial convex polytope", Advances in Mathematics, 35 (3): 236–238, doi:10.1016/0001-8708(80)90050-X, MR 0563925 .
  5. ^ Karu, Kalle (2006), "The cd-index of fans and posets", Compositio Mathematica, 142: 701–718, doi:10.1112/S0010437X06001928, MR 2231198 .

Further reading[edit]