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 f_i i-dimensional faces and f₋₁ = 1. These numbers are arranged into the f-vector of Δ,

${\displaystyle f(\Delta )=(f_{-1},f_{0},\ldots ,f_{d-1}).}$ 

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

For k = 0, 1, …, d, let

${\displaystyle h_{k}=\sum _{i=0}^{k}(-1)^{k-i}{\binom {d-i}{k-i}}f_{i-1}.}$ 

The tuple

${\displaystyle h(\Delta )=(h_{0},h_{1},\ldots ,h_{d})}$ 

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

${\displaystyle \sum _{i=0}^{d}f_{i-1}(t-1)^{d-i}=\sum _{k=0}^{d}h_{k}t^{d-k},}$ 

from which it follows that, for ${\displaystyle i=0,\dotsc ,d}$ ,

${\displaystyle f_{i-1}=\sum _{k=0}^{i}{\binom {d-k}{i-k}}h_{k}.}$ 

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

${\displaystyle P_{R}(t)=\sum _{i=0}^{d}{\frac {f_{i-1}t^{i}}{(1-t)^{i}}}={\frac {h_{0}+h_{1}t+\cdots +h_{d}t^{d}}{(1-t)^{d}}}.}$ 

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.

Recurrence relation

The ${\displaystyle \textstyle h}$ -vector ${\displaystyle (h_{0},h_{1},\dotsc ,h_{d})}$  can be computed from the ${\displaystyle \textstyle f}$ -vector ${\displaystyle (f_{-1},f_{0},\dotsc ,f_{d-1})}$  by using the recurrence relation

${\displaystyle h_{0}^{i}=1,\qquad -1\leq i\leq d}$ 

${\displaystyle h_{i+1}^{i}=f_{i},\qquad -1\leq i\leq d-1}$ 

${\displaystyle h_{k}^{i}=h_{k}^{i-1}-h_{k-1}^{i-1},\qquad 1\leq k\leq i\leq d}$ .

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

${\displaystyle {\begin{matrix}&&&&1&&&\\&&&1&&6&&\\&&1&&&&12&\\&1&&&&&&8\\1&&&&&&&&0\end{matrix}}}$ 

(We set ${\displaystyle f_{d}=0}$  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:

${\displaystyle {\begin{matrix}&&&&1&&&\\&&&1&&6&&\\&&1&&5&&12&\\&1&&4&&7&&8\\1&&3&&3&&1&&0\end{matrix}}}$ 

The entries of the bottom row (apart from the final ${\displaystyle 0}$ ) are the entries of the ${\displaystyle \textstyle h}$ -vector. Hence, the ${\displaystyle \textstyle h}$ -vector of ${\displaystyle \textstyle \Delta }$  is ${\displaystyle \textstyle (1,3,3,1)}$ .

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

${\displaystyle h_{k}=h_{d-k}.}$ 

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:

${\displaystyle h_{k}=\dim _{\mathbb {Q} }\operatorname {IH} ^{2k}(X,\mathbb {Q} )}$ 

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

${\displaystyle rk:P\to \{0,1,\ldots ,n\}}$ 

be the rank function of ${\displaystyle P}$  and let ${\displaystyle P_{S}}$  be the ${\displaystyle S}$ -rank selected subposet, which consists of the elements from ${\displaystyle P}$  whose rank is in ${\displaystyle S}$ :

${\displaystyle P_{S}=\{x\in P:rk(x)\in S\}.}$ 

Then ${\displaystyle \alpha _{P}(S)}$  is the number of the maximal chains in ${\displaystyle P_{S}}$  and the function

${\displaystyle S\mapsto \alpha _{P}(S)}$ 

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

${\displaystyle S\mapsto \beta _{P}(S),\quad \beta _{P}(S)=\sum _{T\subseteq S}(-1)^{|S|-|T|}\alpha _{P}(S)}$ 

is called the flag h-vector of ${\displaystyle P}$ . By the inclusion–exclusion principle,

${\displaystyle \alpha _{P}(S)=\sum _{T\subseteq S}\beta _{P}(T).}$ 

The flag f- and h-vectors of ${\displaystyle P}$  refine the ordinary f- and h-vectors of its order complex ${\displaystyle \Delta (P)}$ :^[8]

${\displaystyle f_{i-1}(\Delta (P))=\sum _{|S|=i}\alpha _{P}(S),\quad h_{i}(\Delta (P))=\sum _{|S|=i}\beta _{P}(S).}$ 

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

${\displaystyle u_{S}=u_{1}\cdots u_{n},\quad u_{i}=a{\text{ for }}i\notin S,u_{i}=b{\text{ for }}i\in S.}$ 

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

${\displaystyle \Psi _{P}(a,b)=\sum _{S}\beta _{P}(S)u_{S}.}$ 

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

${\displaystyle \Psi _{P}(a,a+b)=\sum _{S}\alpha _{P}(S)u_{S}.}$ 

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.^[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

${\displaystyle \Psi _{P}(a,b)=\Phi _{P}(a+b,ab+ba).}$ 

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, S2CID 92984501.
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 [math.CO].
7. Karu, Kalle (2004-08-01). "Hard Lefschetz theorem for nonrational polytopes". Inventiones Mathematicae. 157 (2): 419–447. arXiv:math/0112087. Bibcode:2004InMat.157..419K. doi:10.1007/s00222-004-0358-3. ISSN 1432-1297. S2CID 15896309.
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

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