H-vector explained

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,

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

h _{k} = \sum ^{k} _{i =0} ( -1 ) ^{k-i} \binom{d - i}{k - i}f _{i-1} .

The tuple

h(\Delta) = (h _{0,h} _{1,\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 - \chi(\Delta)) , where \chi(\Delta) is the Euler characteristic of \Delta . The f-vector and the h-vector uniquely determine each other through the linear relation\sum ^{d} _{i =0} f _{i-1} (t -1 ) ^{d-i} = \sum ^{d} _{k =0} h _{k} t ^{d-k} ,

from which it follows that, for

i = 0 , ... c, d ,f _{i-1} = \sum ^{i} _{k =0} \binom{d - k}{i - k} h _{k} .

In particular,

f _{d-1} = h _{0} + h _{1} + ... b + h _{d} . Let R = k[Δ] be the Stanley–Reisner ring of Δ. Then its Hilbert–Poincaré series can be expressed asP _{R} (t) = \sum ^{d} _{i =0} (f _{i-1} t ^{i})/(= ( 1- t) ^{i}) (h _{0+h} _{1t+ … +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

style h -vector (h _{0} , h _{1} , ... c, h _{d} ) can be computed from the style f -vector (f _{-1} , f _{0} , ... c, f _{d-1} ) by using the recurrence relationh ^{i} _{0} = 1 , -1 \le i \le d h ^{i} _{i +1} = f _{i} , -1 \le i \le d -1 h ^{i} _{k} = h ^{i -1} _{k} - h ^{i -1} _{k -1} , 1 \le k \le i \le d .

and finally setting

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

(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:\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

0 ) are the entries of the style h -vector. Hence, the style h -vector of style \Delta is style ( 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

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} = \dim _{Q} \operatorname{IH} ^{2k} (X, 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

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 \left\{ 0 , \ldots, n\right\} , let \alpha _{P(S)} denote the number of chains in P whose ranks constitute the set S . More formally, letrk: P\to\{ 0 , 1 ,\ldots,n\}

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

Then

\alpha _{P(S)} is the number of the maximal chains in P _{S} and the functionS \mapsto \alpha _{P(S)}

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

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

is called the flag h-vector of

P . By the inclusion–exclusion principle,\alpha _{P(S)} = \sum _{T\subseteq} \beta _{P(T).}

The flag f- and h-vectors of

P refine the ordinary f- and h-vectors of its order complex \Delta(P) :^[8] f _{i-1} (\Delta(P)) = \sum _{|S|=i} \alpha _{P(S),} 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, define the corresponding monomial in a and b,u _{S} = u _{1} … u _{n,} u _{i=a} for i\notin S, u _{i=b} for i\in S.

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

\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

\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

\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.

Further reading

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Notes and References

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5. Web site: Amazing: Karim Adiprasito proved the g-conjecture for spheres!. Kalai. Gil. Gil Kalai . 2018-12-25. Combinatorics and more. en. 2019-06-12.
6. Adiprasito. Karim. 2018-12-26. Combinatorial Lefschetz theorems beyond positivity. math.CO. en. 1812.10454v3.
7. Karu. Kalle. 2004-08-01. Hard Lefschetz theorem for nonrational polytopes. Inventiones Mathematicae. en. 157. 2. 419–447. 10.1007/s00222-004-0358-3. 1432-1297. math/0112087. 2004InMat.157..419K. 15896309.
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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.
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