# 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**of a ranked poset. For Eulerian posets, it can be more concisely expressed by means of a noncommutative polynomial in two variables called the

*h*-vector**.**

*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,

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

*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

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

*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 *α*_{P}(*S*) and *β*_{P}(*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 Φ_{P}(*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

- McMullen, Peter (1971), "The numbers of faces of simplicial polytopes",
*Israel Journal of Mathematics*,**9**(4): 559–570, doi:10.1007/BF02771471, MR 0278183. - 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. - 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. - 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. - Kalai, Gil (2018-12-25). "Amazing: Karim Adiprasito proved the g-conjecture for spheres!".
*Combinatorics and more*. Retrieved 2019-06-12. - Adiprasito, Karim (2018-12-26). "Combinatorial Lefschetz theorems beyond positivity". arXiv:1812.10454v3. Bibcode:2018arXiv181210454A. Cite journal requires
`|journal=`

(help) - 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. - Stanley, Richard (1979), "Balanced Cohen-Macaulay Complexes",
*Transactions of the American Mathematical Society*,**249**(1): 139–157, doi:10.2307/1998915, JSTOR 1998915. - 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. - Karu, Kalle (2006), "The
*cd*-index of fans and posets",*Compositio Mathematica*,**142**(3): 701–718, doi:10.1112/S0010437X06001928, MR 2231198.

## Further reading

- Stanley, Richard (1996),
*Combinatorics and commutative algebra*, Progress in Mathematics,**41**(2nd ed.), Boston, MA: Birkhäuser Boston, Inc., ISBN 0-8176-3836-9. - Stanley, Richard (1997),
*Enumerative Combinatorics*,**1**, Cambridge University Press, ISBN 0-521-55309-1.