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 and proved by Lou Billera and Carl W. Lee and Richard Stanley (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. 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.

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 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. 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. 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_{i=0}^k (-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_{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 i = 0, \dotsc, d , f_{i-1} = \sum_{k=0}^i \binom{d-k}{i-k} h_{k}. In particular, 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 P_{R}(t)=\sum_{i=0}^{d}\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. h^{i}_{0} = 1, \qquad -1 \le i \le d h^{i}_{i+1} = f_{i}, \qquad -1 \le i \le d-1 h^{i}_{k} = h^{i-1}_{k} - h^{i-1}_{k-1}, \qquad 1 \le k \le i \le d . and finally setting \textstyle h_{k} = h^{d}_{k} for \textstyle 0 \le k \le d . For small examples, one can use this method to compute \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 \textstyle \Delta of an octahedron. The \textstyle f -vector of \textstyle \Delta is \textstyle (1, 6, 12, 8) . To compute the \textstyle h -vector of \Delta , construct a triangular array by first writing d+2 \textstyle 1 s down the left edge and the \textstyle 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 \textstyle h -vector. Hence, the \textstyle h -vector of \textstyle \Delta is \textstyle (1, 3, 3, 1) . 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_{\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. 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 function S \mapsto \alpha_P(S) is called the flag f-vector of P. The function S \mapsto \beta_P(S), \quad \beta_P(S) = \sum_{T \subseteq S} (-1)^ \alpha_P(S) is called the flag h-vector of P . By the inclusion–exclusion principle, \alpha_P(S) = \sum_{T\subseteq S}\beta_P(T). The flag f- and h-vectors of P refine the ordinary f- and h-vectors of its order complex \Delta(P) : 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 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 \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 \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. 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. The combinatorial meaning of these non-negative coefficients (an answer to the question "what do they count?") remains unclear. ==References== ==Further reading== . .

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