H-vector
Encyclopedia
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 and proved by Lou Billera and Carl W. Lee and Richard Stanley
(g-theorem). The definition of h-vector applies to arbitrary abstract simplicial complex
es. The g-conjecture states that for simplicial spheres, all possible h-vectors occur already among the h-vectors of the boundaries of convex simplicial polytopes.
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 poset
s, 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.
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. 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
Algebraic combinatorics
Algebraic combinatorics is an area of mathematics that employs methods of abstract algebra, notably group theory and representation theory, in various combinatorial contexts and, conversely, applies combinatorial techniques to problems in algebra....
, the h-vector of a simplicial polytope
Simplicial polytope
In geometry, a simplicial polytope is a d-polytope whose facets are all simplices.For example, a simplicial polyhedron contains only triangular faces and corresponds via Steinitz's theorem to a maximal planar graph....
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 and proved by Lou Billera and Carl W. Lee and Richard Stanley
Richard P. Stanley
Richard Peter Stanley is the Norman Levinson Professor of Applied Mathematics at the Massachusetts Institute of Technology, in Cambridge, Massachusetts. He received his Ph.D. at Harvard University in 1971 under the supervision of Gian-Carlo Rota...
(g-theorem). The definition of h-vector applies to arbitrary abstract simplicial complex
Abstract simplicial complex
In mathematics, an abstract simplicial complex is a purely combinatorial description of the geometric notion of a simplicial complex, consisting of a family of finite sets closed under the operation of taking subsets...
es. The g-conjecture states that for simplicial spheres, all possible h-vectors occur already among the h-vectors of the boundaries of convex simplicial polytopes.
Stanley introduced a generalization of the h-vector, the toric h-vector, which is defined for an arbitrary ranked poset
Ranked poset
In mathematics, a ranked partially ordered set - or poset - may be either:* a graded poset, or* a poset that has the property that for every element x, all maximal chains among those with x as greatest element have the same finite length, or...
, and proved that for the class of Eulerian poset
Eulerian poset
In combinatorial mathematics, an Eulerian poset is a graded poset in which every nontrivial interval has the same number of elements of even rank as of odd rank. An Eulerian poset which is a lattice is an Eulerian lattice. These objects are named after Leonhard Euler...
s, 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 complexAbstract simplicial complex
In mathematics, an abstract simplicial complex is a purely combinatorial description of the geometric notion of a simplicial complex, consisting of a family of finite sets closed under the operation of taking subsets...
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. 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
Hilbert–Poincaré series
In mathematics, and in particular in the field of algebra, a Hilbert–Poincaré series , named after David Hilbert and Henri Poincaré, is an adaptation of the notion of dimension to the context of graded algebraic structures...
can be expressed as
-
This motivates the definition of the h-vector of a finitely generatedFinitely generated algebraIn mathematics, a finitely generated algebra is an associative algebra A over a field K where there exists a finite set of elements a1,…,an of A such that every element of A can be expressed as a polynomial in a1,…,an, with coefficients in K...
positively graded algebraGraded algebraIn mathematics, in particular abstract algebra, a graded algebra is an algebra over a field with an extra piece of structure, known as a gradation ....
of Krull dimensionKrull dimensionIn commutative algebra, the Krull dimension of a ring R, named after Wolfgang Krull , is the supremum of the number of strict inclusions in a chain of prime ideals. The Krull dimension need not be finite even for a Noetherian ring....
d as the numerator of its Hilbert–Poincaré series written with the denominator (1 − t)d.
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 posetEulerian posetIn combinatorial mathematics, an Eulerian poset is a graded poset in which every nontrivial interval has the same number of elements of even rank as of odd rank. An Eulerian poset which is a lattice is an Eulerian lattice. These objects are named after Leonhard Euler...
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 cohomologyIntersection cohomologyIn topology, a branch of mathematics, intersection homology is an analogue of singular homology especially well-suited for the study of singular spaces, discovered by Mark Goresky and Robert MacPherson in the fall of 1974 and developed by them over the next few years.Intersection cohomology was...
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 cohomologyIntersection cohomologyIn topology, a branch of mathematics, intersection homology is an analogue of singular homology especially well-suited for the study of singular spaces, discovered by Mark Goresky and Robert MacPherson in the fall of 1974 and developed by them over the next few years.Intersection cohomology was...
groups of X:
(the odd intersection cohomologyIntersection cohomologyIn topology, a branch of mathematics, intersection homology is an analogue of singular homology especially well-suited for the study of singular spaces, discovered by Mark Goresky and Robert MacPherson in the fall of 1974 and developed by them over the next few years.Intersection cohomology was...
groups of X are all zero). The Dehn–Sommerville equations are a manifestation of the Poincaré dualityPoincaré dualityIn mathematics, the Poincaré duality theorem named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds...
in the intersection cohomology of X.
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 posetGraded posetIn mathematics, in the branch of combinatorics, a graded poset, sometimes called a ranked poset , is a partially ordered set P equipped with a rank function ρ from P to N compatible with the ordering such that whenever y covers x, then...
of rank n − 1, so that each maximal chain in P has length n. For any S, a subset of {1,…,n}, let αP(S) denote the number of chains in P whose ranks constitute the set S. More formally, let
be the rank function of P and let PS be the S-rank selected subposet, which consists of the elements from P whose rank is in S:
Then αP(S) is the number of the maximal chains in P(S) and the function
is called the flag f-vector of P. The function
-
is called the flag
The flag f- and h-vectors of P refine the ordinary f- and h-vectors of its order complex Δ(P):
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,
-
Then the noncommutative generating function for the flag
From the relation between αP(S) and βP(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 posetEulerian posetIn combinatorial mathematics, an Eulerian poset is a graded poset in which every nontrivial interval has the same number of elements of even rank as of odd rank. An Eulerian poset which is a lattice is an Eulerian lattice. These objects are named after Leonhard Euler...
P. 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. The combinatorial meaning of these non-negative coefficients (an answer to the question "what do they count?") remains unclear. -
