H-vector

Summary

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.[citation needed] 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

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Let Δ be an abstract simplicial complex of dimension d − 1 with fi 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. In particular,  ,  , and  , where   is the Euler characteristic of  . The f-vector and the h-vector uniquely determine each other through the linear relation

 

from which it follows that, for  ,

 

In particular,  . 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.

Recurrence relation

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The  -vector   can be computed from the  -vector   by using the recurrence relation

 
 
 .

and finally setting   for  . For small examples, one can use this method to compute  -vectors quickly by hand by recursively filling the entries of an array similar to Pascal's triangle. For example, consider the boundary complex   of an octahedron. The  -vector of   is  . To compute the  -vector of  , construct a triangular array by first writing    s down the left edge and the  -vector down the right edge.

 

(We set   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  ) are the entries of the  -vector. Hence, the  -vector of   is  .

Toric h-vector

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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 yP, 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

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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 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 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

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  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

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  • 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.