h-vector

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,

: $f(\backslash Delta)=(f\_\{-1\},f\_0,\backslash ldots,f\_\{d-1\}).$

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

For k = 0, 1, …, d, let

: $h\_k\; =\; \backslash sum\_\{i=0\}^k\; (-1)^\{k-i\}\backslash binom\{d-i\}\{k-i\}f\_\{i-1\}.$

The tuple

: $h(\backslash Delta)=(h\_0,h\_1,\backslash ldots,h\_d)$

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

: $\backslash sum\_\{i=0\}^\{d\}f\_\{i-1\}(t-1)^\{d-i\}=\; \backslash sum\_\{k=0\}^\{d\}h\_\{k\}t^\{d-k\},$

from which it follows that, for $i\; =\; 0,\; \backslash dotsc,\; d$,

:$f\_\{i-1\}\; =\; \backslash sum\_\{k=0\}^i\; \backslash binom\{d-k\}\{i-k\}\; h\_\{k\}.$

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

: $P\_\{R\}(t)=\backslash sum\_\{i=0\}^\{d\}\backslash frac\{f\_\{i-1\}t^i\}\{(1-t)^\{i\}\}=$

\frac{h_0+h_1t+\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.

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

:$h^\{i\}\_\{0\}\; =\; 1,\; \backslash qquad\; -1\; \backslash le\; i\; \backslash le\; d$

:$h^\{i\}\_\{i+1\}\; =\; f\_\{i\},\; \backslash qquad\; -1\; \backslash le\; i\; \backslash le\; d-1$

:$h^\{i\}\_\{k\}\; =\; h^\{i-1\}\_\{k\}\; -\; h^\{i-1\}\_\{k-1\},\; \backslash qquad\; 1\; \backslash le\; k\; \backslash le\; i\; \backslash le\; d$.

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

:

(We set $f\_\{d\}\; =\; 0$ 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:

:

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

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:

: $h\_k\; =\; \backslash dim\_\{\backslash mathbb\{Q\}\}\; \backslash operatorname\{IH\}^\{2k\}(X,\backslash mathbb\{Q\})$

(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.{{Cite journal|last=Karu|first=Kalle|date=2004-08-01|title=Hard Lefschetz theorem for nonrational polytopes|journal=Inventiones Mathematicae|language=en|volume=157|issue=2|pages=419–447|doi=10.1007/s00222-004-0358-3|issn=1432-1297|arxiv=math/0112087|bibcode=2004InMat.157..419K|s2cid=15896309}}

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

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

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 $S$:

:$P\_S=\backslash \{x\backslash in\; P:\; rk(x)\backslash in\; S\backslash \}.$

Then $\backslash alpha\_P(S)$ is the number of the maximal chains in $P\_S$ and the function

: $S\; \backslash mapsto\; \backslash alpha\_P(S)$

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

: $S\; \backslash mapsto\; \backslash beta\_P(S),\; \backslash quad$

\beta_P(S) = \sum_{T \subseteq S} (-1)^

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

: $\backslash alpha\_P(S)\; =\; \backslash sum\_\{T\backslash subseteq\; S\}\backslash beta\_P(T).$

The flag f- and h-vectors of $P$ refine the ordinary f- and h-vectors of its order complex $\backslash Delta(P)$:{{citation

| last = Stanley | first = Richard

| doi = 10.2307/1998915

| journal = Transactions of the American Mathematical Society

| pages = 139{{ndash}}157

| title = Balanced Cohen-Macaulay Complexes

| volume = 249

| number = 1

| year = 1979| jstor = 1998915

| doi-access = free

}}.

:$f\_\{i-1\}(\backslash Delta(P))\; =\; \backslash sum\_\{|S|=i\}\; \backslash alpha\_P(S),\; \backslash quad$

h_{i}(\Delta(P)) = \sum_{|S|=i} \beta_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,

: $u\_S\; =\; u\_1\; \backslash cdots\; u\_n,\; \backslash quad$

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

: $\backslash Psi\_P(a,b)\; =\; \backslash sum\_\{S\}\; \backslash beta\_P(S)\; u\_\{S\}.$

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

: $\backslash Psi\_P(a,a+b)\; =\; \backslash sum\_\{S\}\; \backslash alpha\_P(S)\; u\_\{S\}.$

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.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.

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

: $\backslash Psi\_P(a,b)\; =\; \backslash Phi\_P(a+b,\; ab+ba).$

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.{{citation

| last = Karu | first = Kalle

| mr = 2231198

| doi = 10.1112/S0010437X06001928

| journal = Compositio Mathematica

| pages = 701{{ndash}}718

| title = The cd-index of fans and posets

| volume = 142

| issue = 3

| year = 2006| doi-access = free

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

{{reflist}}

• {{citation

| last = Stanley | first = Richard | author-link = Richard P. Stanley

| edition = 2nd

| isbn = 0-8176-3836-9

| location = Boston, MA

| publisher = Birkhäuser Boston, Inc.

| series = Progress in Mathematics

| title = Combinatorics and commutative algebra

| volume = 41

| year = 1996}}.

• {{citation

| last = Stanley | first = Richard | author-link = Richard P. Stanley

| isbn = 0-521-55309-1

| publisher = Cambridge University Press

| title = Enumerative Combinatorics

| url = http://www-math.mit.edu/~rstan/ec/

| volume = 1

| year = 1997}}.

Category:Algebraic combinatorics

Category:Polyhedral combinatorics