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 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 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[5][6].

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 fi i-dimensional faces and f1 = 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

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 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. Kalle Karu proved that the toric h-vector of a polytope is unimodal, regardless of whether the polytope is rational or not[7].

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 be a finite graded poset of rank n, so that each maximal chain in has length n. For any , a subset of , let denote the number of chains in whose ranks constitute the set . More formally, let

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

Then is the number of the maximal chains in and the function

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

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

The flag f- and h-vectors of refine the ordinary f- and h-vectors of its order complex :[8]

The flag h-vector of can be displayed via a polynomial in noncommutative variables a and b. For any subset 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. [9]


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

References

  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 0551759.
  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. Kalai, Gil (2018-12-25). "Amazing: Karim Adiprasito proved the g-conjecture for spheres!". Combinatorics and more. Retrieved 2019-06-12.
  6. Adiprasito, Karim (2018-12-26). "Combinatorial Lefschetz theorems beyond positivity". arXiv:1812.10454v3. Bibcode:2018arXiv181210454A. Cite journal requires |journal= (help)
  7. Karu, Kalle (2004-08-01). "Hard Lefschetz theorem for nonrational polytopes". Inventiones Mathematicae. 157 (2): 419–447. arXiv:math/0112087. doi:10.1007/s00222-004-0358-3. ISSN 1432-1297.
  8. Stanley, Richard (1979), "Balanced Cohen-Macaulay Complexes", Transactions of the American Mathematical Society, 249 (1): 139–157, doi:10.2307/1998915, JSTOR 1998915.
  9. 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.
  10. Karu, Kalle (2006), "The cd-index of fans and posets", Compositio Mathematica, 142 (3): 701–718, doi:10.1112/S0010437X06001928, MR 2231198.

Further reading

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